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LIBRARY OF CONGRESS. 



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^ PRESENTED BY 

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UNITED STATES OF AMERICA. 





ELEMENTS 




oy 



AL PHILOSOPHY 





W. H. C. BARTLETT, LL.D., 

" pfe»FESSOE OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE UNITED STATES 

. ■" MILITARY ACADEMY AT WEST POINT, 

, > AUTHOR OP 

- /—*C " ELEMENTS OF MECHANICS," " ACOUSTICS," " OPTICS," A ST> 

"ANALYTICAL MECHANICS." 



/y£r^ 



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IV.— SPHERICAL ASTRONOMY 



FIFfH EDITION, REVISED AND CORRECTED. 






NEW YOEK: 



A. S. BARNES & CCL 111 & 113 WILLIAM ST., COR. OF JOHN 






SOLD BY BOOKSELLERS GENERALLY, THROUGHOUT THE UNITED STATES. 

1871. 
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Valuable f orb by Leaiiu Altai ? Ci i 

TN the * ■ 

HIGHER MATHEMATICS. 

» «g»^» 

W. M* G. BARTLETT, LL.D., 

^V-o/. of Nat. & LJxp. Philos. in the U. S. Military Academy, West Point. 
BARTLETT'S SYNTHETIC MECHANICS. 
Elements of Mechanics, embracing Mathematical formulae for observing and calculating 
the action of Forces upon Bodies— the source of all physical phenomena. 

BARTLETT'S analytical mechanics. 
For more advanced students than the preceding, the subjects being discussed Analytically, 
by the aid of Calculus. 

BARTLETT'S ACOUSTICS AND OPTICS. 
Treating Sound and Light as disturbances of the normal Equilibrium of an analogous char- 
acter, and to be considered under the same general laws. 



B4RTiiETT'S ASTRONOMY. 

I] 

18784 



Spherical Astronomy in its relations to Celestial Mechanics, with full applications to the 
current wants of Navigation, Geography, and Chronology. 



A. E. CHURCH, LL.D., 

Prof. Mathematics in the United Stales Military Academy, West Point. 

CHURCH'S ANALYTICAL GEOMETRY. 
Elements of Analytical Geometry, preserving the true spirit of Analysis, and rendering the 
whole subject attractive and easily acquired. 

CHURCH'S CALCULUS. 
Elements of the Differential and Integral Calculus, with the Calculus of Variations. 

CHURCH'S DESCRIPTIVE GEOMETRY. 
Elements of Descriptive Geometry, with its applications to Spherical Projections, Shades 
and Shadows, Perspective and Isometric Projections. 2 vols. ; Text and Plates respectively. 



EDWARD M. COURTENAY, LL.D., 

Late Prof. Mathematics in the University of Virginia. 

COURTENAYS CALCULUS. 
A treatise on the Differentia] and Integral Calculus, and on the Calculus of Variations. 



CHAS. W. HACKLEY, S.T.D., 

Late Prof, of Mathematics and Astronomy in Columbia College. 
HACKLEYS TRIGONOMETRY. 
A treatise on Trigonometry, Plane and Spherical, with its application to Navigation and 
Surveying, Nautical and Practical Astronomy and Geodesy, with Logarithmic, Trigonomet- 
rical, and Nautical Tables J 

DAVIES & PECK, 

Department of Mathematics, Columbia College. 
MATHEMATICAL DICTIONARY 
And Cyclopedia of Mathematical Science, comprising Definitions of all the terms employed 
in Mathematics— an analysis of each branch, and of the whole as forming a single science. 

CHARLES DAVIES, LL.D., 

Late of the United Slates Military Academy and of Columbia College. 

A COMPLETE COURSE IN MATHEMATICS. 

See A. S. Babnes & Co.'s Descriptive Catalogue. 



Entered, according to Act of Congress, in the year 1859, by 
W. H. C. BARTLETT, 
In the Clerk's Office of the District Court of the United States for the Southern District of New York. 
B'S A'MY. 






'/ 



PREFACE. 



The work here offered to the public was undertaken by its 
author to supply a want long felt in his own department of 
instruction in the Military Academy at "West Point. Its aim 
is to present a concise course of Spherical Astronomy in its 
relationship to Celestial Mechanics, of which it is the offspring. 
The solar and stellar systems are, therefore, assumed and de- 
scribed as necessary facts, arising from the detached condition 
of the bodies which compose them and the laws of universal 
gravitation. The consequences from these systems, to a spec- 
tator on the earth, are then deduced, and their entire coinci- 
dence with the celestial phenomena, as they arise spontane- 
ously, is relied upon as full and sufficient justification for the 
assumption, and as proof that the systems are true. This 
forms the first part of the subject. A general account of the 
methods by which the future condition and aspects of the 
heavens are predicted follows, and the more important appli- 
cations to the current wants of Navigation, Geography, and 
Chronology, conclude the volume. 



IT 



PREFACE. 

In the description and discussions of instruments, those only 
have been selected which are best suited to convey a full view 
of the whole theory and practice of Astronomical Measure- 
ments. 

The author would acknowledge his obligation to Sir John 
Herschel, Professor Challis, Mr. Maddy, Mr. .Francis Bailey 
Mr. De Morgan, Mr. Woolhouse, M. Francceur, M. De Launay 
and M. Briot, whose works have been constantly before him. 




] <lj 












3/ /? *&04r&Jte-c s^ 



CONTENTS. 



... 

Introductory Remarks * 1 

Solar System 2 

potion 6 

Parallactic Motion 6 

Celestial Sphere 7 

Shape of the Earth 8 

Diurnal Motion 9 

Definitions 10 

Instruments IE 

Proportions of Land and Water — Atmosphere 13, 

Refraction 14 

Parallelism of the Earth's Axis, and Uniformity of the Earth's Diurnal Motion. . . . 16 

Upper and Lower Diurnal Arcs — Circumpolar Bodies 18 

Terrestrial Latitude and Longitude 20 

Figure and Dimensions of the Earth 21 

Geocentric Parallax 24 

Augmented Horizontal Diameters 2$ 

Distances and Dimensions of the Heavenly Bodies 29 

Ecliptic 30 

Precession and Nutation 37 

Sidereal Time 40 

Earth's Orbit 41 

Mean Solar Time 46 

Aberration .".... 50 

Heliocentric Parallax 53 

The Seasons 56 

Trade-Winds 59 

Terrestrial Magnetism 62 

Tides 66 

Twilight 72 

The Sun 76 

Planets 84 

Elements of the Planets 35 

Dimensions and Distances of Planets 90 



\\ 



\ 



PAGB 



vi CONTENTS. 

Inferior Planets — Superior Pknets *.. « >k . . . . .'\. '.. ..... . 90 

Synodic Bevolutions — Geocentric Motions .»-.-; 91 

Direct and Betrograde Motions — Station^.:, ...s *. ...io.V. . .}.% 92 

Phases of the Planets ' 93 

Transits — Occupations 95 

Masses and Densities of the Planets 97 

Mercury 99 

Venus 100 

Mars— Planetoids . . . , 102 

Jupiter — Saturn 104 

Uranus — Neptune 108 

Secondary Bodies 109 

The Moon— Lunar Orbit 11C 

Disturbing Forces 113 

Librations 115 

Lunar Periods 116 

Lunar Phases -. 117 

Eclipses of the Sun and Moon 118 

Moon's Eelative Geocentric Orbit 123 

Ecliptic Limits 124 

Number of Eclipses 125 

The Saros 126 

Physical Constitutior oi the Moon 127 

Satellites of Jupiter 130 

Progressive Motion of Light 134 

Satellites of Saturn 134 

Satellites of Uranus 136 

Satellites of Neptune 137 

Comets 137 

Elements of the Orbits of the Permanent Comets 140 

Stars 144 

Elements of Stellar Orbits 156 

Proper Motion of the Stars and of the Sun 158 

Nebulas 159 

Zodiacal Light 162 

Aerolites — Meteors 163 

Ephemerides 165 

Catalogue of Stars 170 

Applications 174 

Time of Conjunction and of Opposition 174 

Angle of Position 175 

Projection of a Solar Eclipse 175 

Projection of a Lunar Eclipse 183 

Time of Day 184 

Azimuths 188 

Meridian Passages 191 

Eeduction to the Meridian 192 

Terrestrial Latitude 195 

Terrestrial Longitude 202 

Calendar 229 



CONTENTS Vii 

PAGE 

Appejjdix I. — Elements of the Principal P'anets ... 238 

Appendix II. — Astronomical Instrumc at? 237 

Clock and Chronometer 237 

Vernier ." 243 

Micrometer « 245 

Level 250 

Eeading Microscopes ' 252 

Transit 255 

Collimating Telescope 266 

Vertical Collimator 267 

Collimating Eye-piece 268 

Mural Circle. 269 

Altitude and Azimuth Instrument 276 

Equatorial .' 282 

Heliometer 293 

Sextant 294 

Artificial Horizon '. . . 298 

Principle of Eepetition 300 

Eeflecting Circle 301 

Appendix III. — Atmospheric Eefraction 305 

Appendix IV. — Shape and Dimensions of the Earth 310 

Appendix V.— The Earth's Orbit 313 

Appendix VI. — Planets' Elements 316 

Appendix VII.— Planets' Elements . 317 

Appendix VIII.— Planets' Elements 318 

Appendix IX.— Planets' Elements 319 

Appendix X. — Geocentric Motion 331 

Appendix XI. — Mr. Woolhouse on Eclipses, &c » 333 

Appendix XII. — Equation of Equal Altitudes 424 

Appendix XIII. — Correction for Difference of Eefraction , 42fi 



TABLES. 

Table I. — Mr. Ivory's Mean Eefractions, with the Logarithms and their Differ- 
ences annexed 42? 

Table II. — Mr. Ivory's Eefractions continued: showing the Logarithms of the 
corrections, on account of the state of the Thermometer and 
Barometer 430 

Table III. — Mr. Ivory's Eefractions continued : showing the further quantities 
by which the Eefraction at low altitudes is to be corrected, on 
account of the state of the Thermometer and Barometer 431 

Table IV.— For the Equation of Equal Altitudes of the Sun 432 

Table V. — For the Eeduction to the Meridian: showing the valae of 

A^-^L? ; '. 449 

sin V, 

Table VI. — For the second part of the Eeduction to the Meridian : showing the 

2 sin 4 h P 
value of B = — : „— 450 

sin V 

Trigonometrical Formula 451 



The Greek Alphabet is here inserted to aid those who are not already 
amiliar with it in reading the parts of the text in which its letters occur : 



Letters. 


Names* 


Letters. 


Name*. 


A a 


Alpha 


N v 


Nu 


B /3g 


Beta 


s'i 


Xi 


r yf 


Gamma 


O o 


Gmicroi 


A 5 


Delta 


n TXit 


Pi 


E £ 


Epsilon 


p H 


Rho 


z?£ 


Zeta 


2 (fs 


Sigma 


Ht, 


Eta 


T «r7 


Tau 


&0 


Theta 


T u 


Upsilon 


I i 


Iota 


<l> © 


Phi 


K x 


Kappa 


xx 


Chi 


A X 


Lambda 


Y + 


Psi 


M p 


Mu 


n a 


Omega 



Conventional Signs used in Astronomy. 

L, for mean longitude, 
M, — mean anomaly, 
V, — true anomaly, 
/x, — mean daily sidereal motion, 
r, — radius vector, 
9, — - angle of eccentricity, 
cr, — longitude of perihelion, 
a, — right ascension, 
6, — declination, 

A, — logarithm of distance from the earth, 
I, — heliocentric longitude, 
by — heliocentric latitude, 
X, — geocentric longitude, 
/3, — geocentric latitude, 
S, — longitude of ascending node, 
i, — inclination of orbit to the ecliptic, 
Wy — angular distance from perihelion to node, 
w, — distance from node, or argument for latitude. 




ASTRONOMY. 



ASTRONOMY. 



§ 1. The science which treats of the heavenly bodies is called Asiron- 
®my. It is divided into Physical and Spherical Astronomy. 

§ 2. Physical Astronomy is a system of Mechanics, in which the forces 
are universal gravitation and inertia, and the objects the gigantic masses 
that move through indefinite space. It treats of the physical conditions 
of the heavenly bodies, their mutual actions on each other, and explains 
the causes of the celestial phenomena. 

§ 3. Spherical Astronomy is mainly concerned with the appearances, 
magnitudes, motions, arrangements, and distances of the heavenly bodies ; 
and seeks to apply the deductions from these to the practical wants of 
society. It is a science of observation, and its principal means of investi- 
gation are Optical and. Mathematical Instruments. This branch of As- 
tronomy will form the subject of the present volume. 

§ 4. No subject calls more strongly upon the student to abandon first 
impressions than Astronomy. All its conclusions are in striking contra- 
diction to those of superficial observation, and to what appears, at first 
view, the most positive evidence of the senses. 

§ 5. Every student approaches it for the first time with a firm belief 
that he lives on something fixed, and, abating the inequalities of hill and 
"•alley, that this something is a flat surface of indefinite extent, composed 
of land and water ; and that the blue firmament which he sees around 
and above him in the distance is a stationary vault, upon the surface of 
which appear to be placed all objects out of contact with the ground. 

§ 6. The Earth on which he stands is divested by Astronomy of its 
flattened shape and of its character of fixidity, and is shown to be a 
globular body turning swiftly about its centre, and moving onward through 
space with great rapidity. It teaches him that his vault has no existence 



2 ASTRONOMY. 

in fact, and is but a mere illusion which comes from looking through the 
indefinite space, extended without limit, in which he is movino-. 

§ 7. Were the Earth reduced to a mere point, and a spectator placed 
upon it, he would see around him at one view all the bodies which make 
up the visible universe ; and in the absence of any means of judging of 
their distances from him, would refer them in the direction in which they 
were seen from his station, to the concave surface of an imaginary sphere, 
having its centre at his eve and its surface at some vast and indefinite 
distance. 

SOLAR SYSTEM. 

§ 8. A little observation would lead him to conclude that by far the 
greater number of these bodies appear fixed while the rest seem ever on 
the move, continually shifting their positions with respect to those which 
appear fixed, and to each other. The former are called Fixed Stars ; 
the latter compose what is called the Solar System, a group of bodies 
from which the fixed stars are so remote as to produce upon it no appre- 
ciable influence. 

§ 9. All bodies attract one another with intensities which are propor- 
tional to the quantity of the attracting masses directly, and to the squares 
ef the distances inversely, Analyt. Mech., § 205. 

§ 10. Bodies resist by their inertia all change in their actual state of 
motion ; this -resistance is exerted simultaneously with the change, and is 
always equal in intensity, and contrary in direction, to the force which 
produces it. j 

§ 11. The bodies of the solar system have motions that carry them in 
directions oblique to the lines along which their mutual attractions are 
exerted. The attractive forces draw them aside from these directions ; 
inertia resists by an equal and contrary reaction ; and the bodies are 
forced into curvilinear paths, and made to revolve about the centre of 
inertia of the whole. 

§ 12. Thus, the antagonistic forces of gravitation and of inertia are the 
-simple but efficient causes which keep the bodies of the solar system to- 
gether as a single group, and impress upon it a character of stability and 
perpetuity. But for the force of gravitation the bodies would separate 
more and more, and winder through endless space ; and but for the force cf 
inertia, that of gravitation would pile them together in one confused mass. 

§ 13. The force of gravitation increases rapidly with a diminution, and 
decreases as rapidly with an augmentation, of distance. Those bodies 
which are nearest exert, therefore, the greatest influence upon one another's 



SOLAR SYSTEM. 3 

motions. Bodies composing an insulated group may perform their evolu- 
tions among e;ich other undisturbed by the action of those without, pro- 
vided the distances of the latter be very great in compaiison to those 
which separate the individuals of the group, 

§ 14. This is a characteristic of the solar system. Its own dimensions, 
vast as they are when expressed in terms of any linear unit with which 
we are familiar, are utterly insignificant when compared with its distance 
from the fixed stars. Each of the latter, by virtue of this relatively great 
distance, acting upon all the bodies of the system equally and in parallel 
directions, the effect of the whole can only be to move the group collec- 
tively through space, , . . , 

§ 15. The same thing takes place upon a smaller scale within the solar 
system itself. Some oi' its members are so close together, and at the same 
time so far removed from the others, as to be forced to revolve about one 
anotl^-, while the combined action of the rest carries them as a sub-group, 
so to speak, about the centre of inertia of the whole. 

§ 16. The ^ass of the sun so far exceeds the sum of the masses of all 
the other bodies of the sy^m, as to throw the centre of ineina of the 
whole group within the boundary of its ow^ volume ; and although the 
centre of the sun actually revolves about this point, yet its mocion be- 
comes so small, when viewed from the distance of the earth, that it is in- 
sensible except through the medium of the most refined instruments. All 
the other bodies are, therefore, said to revolve about the sun as a centre, 
and it is from this fact, and the controlling influence which this latter 
body exerts over the motions of all the others, that the system takes its 
name. 

§ 17. The same is true of the sub-groups ; the mass of one of the bodies 
in each being so much greater than the sum of the masses of the rest as 
to cause the latter to revolve approximately about its centre, while this 
centre revolves about the sun. 

§ 18. The path a body describes about another as a principal source of 
attraction, is called an orbit. 

§ 10. Those bodies which describe their orbits about the sin ate called 
primary, and those which desciibe their orbits about the primaries are 
called secondary bodies. These latter are also called Satellites. 

Of the primary bodies there are three distinct classes, differing from 
each other mainly in the shape of their orbits, their densities, and gen- 
eral aspects. 

§ 20. A body subjected to the action of a central force, whose intensity 
Taries as the square of the distance inversely, must desciibe one or other of 



4 ASTRONOMY 

tbe conic sections, depending upon the relation i^tween its velocity and 
the intensity of the central force. The orbits that are known to belong to 
the solar system are ellipses. 

§ 21. Those primaries which move in elliptical orbits of small eccentri- 
cities are called Planets. Those primaries having orbits of great eccentri- 
cities are called Comets. Comets are also distinguished from planets in 
having a degree of density so low as to give some the appearance more of 
a vapor than of a solid body. 

§ 22. The solar system consists then of the Sun, Planets, Comets, and 
Satellites. Setting out from the sun, the known planets, with their names, 
occur in the following order, viz. : Mercury, Venus, the Earth, Mars, 
then a class called the Planetoids, of which ninety-one are known at the 
present time, Jupiter, Saturn, Uranus, and Neptune. Bee Plate I., Fig. I. 

To these must be added a multitude of much smaller bodies of the 
nature of planetoids, whose existence is inferred from the fact thar some 
of their number make their way now and then to the earth's surface under 
tbe name of meteors. 4} 

§ 23. It would be utterly impossible to gi#e within the narrow limits of 
an octavo page a graphical representation of the relative dimensions of the 
solar system ; and to aid the conceptions of tbe student, Sir John Herschel 
has instituted the following illustration, viz. : On any well-levelled field 
place a globe two feet in diameter ; this will represent the sun ; Mercury 
will be represented by a grain of mustard-seed on the circumference of a 
circle 164 feet in diameter for its orbit ; Venus a pea on the circumference 
of a circle 284 feet in diameter; the Earth also a pea on the circumference 
of a circle 430 feet in diameter ; Mars a rather large pin's head on th« 
circumference of a circle of 654 feet diameter ; the Planetoids grains of 
sand on circular orbits varying from 1000 to 1200 feet in diameter; 
Jupiter a moderate sized orange on a circumference nearly half a utile 
in diameter ; Saturn a small orange on the circumference of a circle four- 
fifths of a mile in diameter ; Uranus a full sized cherry on the circumfer- 
ence of a circle more than a mile and a half in diameter ; and Neptune 
a good sized plum on the circumference of a circle about two miles and as 
half in diameter. To illustrate the relative motions, Mercury must doserilw 
a portion of its orbit equal in length to its own diameter in 41 seconds : 
Venus in 4 minutes and 14 seconds; the Earth in 7 minutes ; Mars in 4 
minutes and 48 seconds ; Jupiter in 2 hours and 56 minutes ; Saturn in 3 
hours and 13 minutes; Uranus in 2 hours and 16 minutes, and Neptmm 
in 3 hours and 30 minutes. Now conceive the two feet globe to be in- 
creased till its diameter becomes 880,000 English miles, and suppose the 



Plate I 




TO FRONT PtAG-H 4 



SOLAR SYSTEM. 5 

otto bodies and their distances increased in the same proportion ; the re- 
sult will represent the dimensions of the solar system. It will give to 
the earth a diameter of nearly eight thousand miles, a distance from the sun 
equal to 95 millions of miles, and a velocity through space, around the sun, 
of 19 miles a second. 

The orbits, although referred to as circles, are- in fact ellipses, but of ec- 
centricities so small as to justify the substitution for the mere purposes of 
the illustration. 

§ 24. The fixed stars are self-luminous. The sun is regarded as one of 
this class of bodies, and by its greater proximity to the earth, becomes the 
principal source of heat and light to its inhabitants. 

§ 25. The plauets and satellites are opaque uon-luminous bodies, and 
are visible only in consequence of light received from the sun and reflected 
to the earth. 






'0-^tsJ^2^l^C^&6 / sr -e^V '^L^chJ t^Y-z^z^/ 5" ~^~ 



/<*. 



SPHERICAL ASTRONOMY 



MOTION. 

§ 26. Motion signifies the condition of a body, in virtue of which it oc 
cupies successively different places. But we can form no idea of place ex- 
cept by referring it to other places, and these again, to be known, must be 
referred to others, and so without limit ; so that place is, in its very nature, 
entirely relative. Motion is, from its definition, therefore, also relative. 

§ 27. We judge of the rate of motion by the greater or less rapidity 
with which the object possessing it varies its distance from other objects 
assumed as origins. These origins may themselves be in motion, but if 
the circumstances of the spectator be such as to deceive him into the belief 
that they are at rest, he will attribute all change of distance to a motion 
wholly in the object which he refers to them. And this is one of the most 
fruitful sources of the many erroneous notions with which students gener- 
erally commence the study of astronomy. 

§ 28. If two objects be in motion, and they alone occupy the spectator's 
field of view, the effect to him will be the same if he suppose one fixed, and 
attribute the whole of its motion to the other in a contrary direction ; for 
this will not alter the rate by which they approach to or recede from one 
another. 

PARALLACTIC MOTION AND PARALLAX. 

§ 29. The real motion of a spectator gives rise to the appearance of 
motion among surrounding objects which are relatively at rest. Objects 
in front of him seem to separate from one another, those behind appear to 
approach one another, and those directly to the right and left seem to move 
in a direction parallel to his own motion. 

A spectator, for example, travelling over a plain studded w T ith trees or 
other objects will, on fixing his eyes upon a single object without with- 
drawing his attention from the general landscape, see or think he sees the 






CELESTIAL SPHERE. 



7 



latter in rotary motion about that object as a centre ; al' objects between 
it and himself appearing to move backward, or contrary to his own motion, 
and all beyond it, forward or in the direction in which he moves. 

This apparent change in the relative places of objects, arising from a 
shifting of the point of view from which they are seen, is called parallactic 
motion ; and the amount of angular change in the instance of any partic- 
ular object is called the parallax of that object. 

§ 30. Let P be the place of an object, O and S the 
places from which it is seen ; and let its place be referred 
to some point Z\ on the prolongation of the line CS, 
which joins the points of view. The angular change in the 
place of P as seen from C and S will be 

r SP-Z' CP=SP C=the parallax of P. 

That is to say, the parallax of an object is the angle sub- 
tended at the object by the distance between the stations 
from which it is seen. 

Make CP=d; CS=p; the angle Z'SP = Z; the angle S P C=z 
Then from the triangle' CS P, w^Jiave 

%'*™ %;: f h 

NT sin z— ¥, sin Z . (1) 




Whence the parallax increases with an increase of the spectator's change 
of place, with diminution of the object's distance, and also with the approx- 
imation of Z to 90°. 

§ 31. All other things being equal, the parallax will be less as the ob- 
ject's distance is greater; and when the parallax is zero for any arbitrary 

value of Z, the factor j- must be zero, and the change of the spectator's 

place must be utterly insignificant in comparison with the object's distance. 



CELESTIAL SPHERE. 

§ 32. Now, when the heavens are examined it is found that by far the 
greater number of the celestial bodies have no sensible parallax, while 
comparatively a few have. The first are the fixed stars ; and they are so 
called from the fact that they always preserve the same angular distances 
from any assumed point and from each other, from whatever station on the 
earth they are. viewed. The second are bodies of the solar system. 

§ 33. The fixed stars are, therefore, beyond limits at which objects cease 



SPHERICAL ASTRONOMY 



to be sensibly affected by parallax. The great concave of the heavens 
upon which the fixed stars appear to be situated, is called the celestial 
sphere. Not only, therefore, is the longest rectilineal dimension of the 
earth, but also the distance between the points of its orbit about the sun 
most remote from each other — a distance, as we shall see in the sequel, 
equal to one hundred and ninety millions of miles — utterly insignificant 
when expressed in terms of the radius of the celestial sphere as unity. A 
sphere large enough to contain the entire orbit of the earth is a mere point 
in comparison with the vast volume embraced by the celestial sphere. 
The centre of the earth may, therefore, always be regarded as the centre of 
the celestial sphere. 



SHAPE OF THE EARTH. 




§ 34. The earth, being the station from 
which all the other heavenly bodies are 
viewed, is the first to claim attention. It 
has been repeatedly circumnavigated in dif- 
erent directions, and the portions of its sur- 
face visible from elevated positions in the 
midst of extended plains or at sea, always 
appear as circles of which the spectator 
seems to occupy the centre. The apparent 
diameters of these circles, measured by in- 
struments, are smaller in proportion as the 

points of view S are more elevated. The earth is, therefore, ylobular ; 
for to such figures alone belong the property of always presenting to the 
view a circular outline. 

§ 35. By the figure of the earth is meant its general shape without 
regard to the irregularities of surface which form its hills and valleys. 
These are relatively insignificant and are disregarded in speaking of the 
earth's form. They are less in proportion to the entire earth than the 
protuberances and indentations on the surface of a smooth orange are tc 
a large size specimen of that fruit. The earth is an oblate spkeroid r and 
the operations and method of computations by which its precise magni- 
tude and proportions are found, will be given presently. 

The shortest diameter of the earth is called its axis. 









DIURNAL MOTION. 



DIURNAL MOTION. 






§ 36. The boundary of the visible portion of the earth's surface, sup 
perfectly smooth, is called the sensible horizon. The sensible horizon 



2g 





is only seen at sea, or on extended plains. At most localities on land it is 
broken by hills, valleys, and other objects. 

§ 37. The earth conceals from us that portion of space below our sen- 
sible horizon, while all above is exposed to view. It rotates upon its axis, 
and the period required to perform one entire revolution is called a day. 

§ 38. Every spectator is carried about Fig. 4. 

the earth's axis in the circumference of 
a circle, and while the extent of the 
visible portion of space remains un- 
changed, different regions are continu- 
ally passing through the field of view. 
The horizon of a spectator will be ever 
depressing itself below those bodies 
which lie in the region of space towards 
which he is carried by the rotation, and 
elevating itself above those in the oppo- 
site quarter; thus successively bringing into view the former and hiding 
the latter. 

§ 39. The spectator being unconscious of his own motion, concludes, 
from first appearances, that his horizon is at rest, and attributes these 
changes to an actual motion in the objects themselves. Instead of his 
horizon approaching the bodies, he judges the bodies to approach his 
horizon ; and when it passes and hides them, he regards them as having 
sunk below it or set, while those it has just disclosed, and from which it 
is receding, he considers as having come up or risen. 

§ 40. One entire revolution about the axis being completed, the spec- 
tator returns to the place from which he commenced his observations, and 
he begins again to witness the same succession of phenomena and in the 
same order. All the heavenly bodies appear to occupy the same places 
in the concave sky which they did before. 

§ 41. Thus the rotation of the earth about its axis produces the daily 



10 SPHERICAL ASTRONOMY 

rising and setting of the sun — the alternation of day and night ; also the 
rising and setting of the other heavenly bodies, their progress through the 
vault of the heavens, and their return to the same apparent places at short 
and definite intervals. 

§ 42. The apparent motions with reference to the horizon by which 
these daily recurring phenomena are brought about, are called the diurnal 
motions of the heavenly bodies. The real motion is in the horizon, the 
origin of reference ; it is only apparent in the bodies themselves. 

DEFINITIONS. 

§ 43. The axis of the celestial sphere is the axis of the earth produced. 

§ 44. The poles of the earth are the points in which its axis pierces its 
surface. The pole nearest to Greenland is called the north, the other the 
south pole. 

§ 45. The poles of the heavens are the points in which its axis pierces 
the celestial sphere. That above the north pole of the earth is called the 
north, the other the south pole. 

§ 46. The earth's equator is the intersection of the earth's surface by a 
plane through its centre, and perpendicular to its axis. 

§ 47. The equinoctial is the intersection of the surface of the celestial 
sphere by the same plane. 

§ 48. A meridian line is the intersection of the earth's surface by a 
plane through its axis and the place of a spectator. 

§ 49. The celestial meridian is the intersection of the surface of the 
celestial sphere by the same plane. This is often called simply the me- 
ridian of the place. 

§ 50. The poles of the celestial meridian are called the East and West 
points ; that towards which the spectator is moving by his diurnal motion 
being the East, that from which he is receding the West. 

§ 51. The apparent zenith and apparent nadir are the points in which 
a plumb-line produced intersects the celestial sphere : that over head being 
the zenith. 

§ hi. The rational horizon is the intersection of the celestial sphere by 
a plane through the earth's centre and perpendicular to the line of the 
zenith and nadir. The plumb-line being always normal to the earth's sur- 
face, the plane of the ratioual horizon is parallel to the plane tangent to 
the earth's surface at the spectator's place, and these planes intersect the 
celestial sphere sensibly in the same great circle. 

§ 53. The dip of the horizon is the angle which the elements of a 



DEFINITIONS. H 

visual cone, whose vertex is in the eye of the spectator, and whose surface 
is tangent to that of the earth along the sensible horizon, make with the 
tangent plane to the earth at the spectator's place. The dip is greater in 
proportion as the spectator's elevation above the earth is greater. When 
the eye is in the earth's surface, the dip is zero, and the visual cone be- 
comes the tangent plane. This coincidence will always be supposed to 
exist unless the contrary is specially noticed. 

§ 54. The latitude of a place on the earth's suiface is the arc of the 
celestial meridian from the equinoctial to the zenith of the place. It is 
always measured in degrees, minutes, seconds, and thirds. Latitude is 
reckoned north or south ; that reckoned towards the north pole being 
called north latitude, that towards the south pole, south latitude. The 
greatest latitude a place can have is 90°, this being the latitude of the 
poles of the earth. 

§ 55. Parallels of latitude are small circles on the earth's surface par- 
allel to the equator. All places on the same parallel have the same 
latitude. 

§ 56. The longitude of a place on the earth's surface is the arc of the 
equinoctial intercepted between the meridian of the place and that of 
some other place assumed as a first meridian. It is called East or West, 
according as it is reckoned in the direction from the first meridian towards 
its east or west point. For the sake of uniformity, it will, in the text, al 
ways be reckoned in the latter direction. The English estimate longitude 
from the meridian of Greenwich, the French from that of Paris, aud other 
nations from other meridians. In the United States, for most geographical 
purposes, it is estimated from the meridian of Washington. 

§ 57. A vertical circle is the intersection of the celestial sphere by a 
plane through the zenith and nadir. 

The prime vertical is the vertical circle whose plane is perpendicular to 
that of the meridian. 

§ 58. The north and south points are the poles of the prime vertical ; 
that below the north pole being called the north point. 

§ 59. The Azimuth of a body is the angle which a vertical circle 
through the body's centre makes with the meridian. It is measured on 
the horizon, and from the south towards the west, or from the north to- 
wards the west, according as the north or south pole is elevated above the 
horizon. It may vary from 0° to 360°. 

§ 60. The zenith distance of an objec is the angular distance from 
the apparent zenith to the centre of the object, measured on a vertical 
circle. 



12 



SPHERICAL ASTRONOMY. 



§ 61. The altitude of an object is the angular distance from the horizon 
to the object's centre, measured on a vertical circle. 

The azimuth and zenith distance are a species of polar co-ordinates for 
the designating- an object's place in the heavens. By making the azimuth 
vary from zero to 360°, and the zenith distance from zero to 90°, every 
visible point of celestial space may be defined in position. 

§ 62. A declination circle, or hour circle, is the intersection of a plane 
through the axis of the heavens with the celestial sphere. 

§ 63. The declination of an object is the angular distance of its centre 
from the equinoctial, measured on a declination circle. The declination 
may be north or south, and may vary from 0° to 90°. 

§ 64. The polar distance of an object is the angular distance of its 
centre from the celestial pole, measured on a declination circle. 

§ 65. The right ascension of an object is the angle which a declination 
circle through the object's centre makes with a declination circle through 
a certain point on the equinoctial, called the Vernal Equinox. This angle 
is measured upon the equinoctial,- and eastwardly in direction. 

§ 66. The polar distance and right ascension are also a kind of polai 
co-ordinates for defining the places of celestial objects ; for this purpose it 
is only necessary to cause the right ascension to vary from 0° to 360°, and 
the polar distance to vary from 0° to 180°, to reach every point in the 
celestial sphere. 

§ 67. The hour angle of an object is the angle which its hour circle 
makes with the meridian of the place. It is estimated from the meridian 
westwardly, and may vary from to 360°. The hour angle may be em- 
ployed, instead of the right ascension, with the polar distance to define an 
object's place. 

To illustrate, let the plane of 
the paper be that of the meridian ; the 
circle HZ ON its intersection with the 
celestial sphere ; P P' the axis>#^he 
heavens ; P and P' the north ancR^<]h 
poles respectively ; Z and iV the zenith 
and nadir respectively, and the earth a 
mere point at C; then will the circle 
QWQ' E, of which P and P' are the 
poles, be the equinoctial; IT WOE, of 
which Z and JV are the poles, the hori- 
zon ; E and W, the poles of the meridian, will be the east and west points 
respectively; the arc ZQ will be the latitude, ZSjvh vertical circle, 




INSTRUMENTS. 13 

Z S the zenith distance of the object S, A S its altitude, and OWA its 
azimuth ; PS will be its polar distance, D S its declination, Z P S, meas- 
ured by Q D, its hour angle, and if V be the vernal equinox, VD will be 
its right ascension. 

INSTRUMENTS. 

§ 68. Most of the data with which the practical astronomer labor?, 
come from measurements made in the circles just referred to, by means 
of certain astronomical instruments. These instruments are described, 
and their themrjK adjustments, and uses explained, in Appendix II. 
The studejvFslfould study, in connection with short daily lessons of the 
text, #bn*#his point, the Clock. Chronometer, Transit, Mural Circle 
and Azimuth and Altitude Instrument. The others should be taken 
up where referred to, in the order of the text. 

PROPORTIONS OF LAND AND WATER— THE ATMOSPHERE. 

§ 69. To resume the consideration of the earth. About three-fourths 
of its surface are covered with water, and the greatest depth of the sea 
does not probably exceed the greatest elevation of the continents. 

The earth is surrounded by a gaseous envelope, called the atmosphere, 
the actual thfckness of which, were it reduced to a uniform density 
throughout, equal to that at the surface of the sea, would be about 
five miles. But owing to the law which regulates the pressure, density, 
and temperature of elastic bodies, it is much greater than this.- The dif- 
ferent strata, being relieved from the weight of those belosfc them, become 
more expanded in proportion as they are higher, and ^be pace apf the su- 
perior atmospheric limJWfcffu^k result from an equiliwaufct between the 
weight of the terminal stratum and the elastic force of th&t upon which 
it rests. The laws just referre4 to indicate that this limit cannot be much 
higher than 80 miles. **»*» • - 

§ 70. The atmosphere is not perfectly transparent. The sun illumines 
its particles ; these scatter by reflection the light they receive-, particularly 
the blue, in all directions, and produce that general illumination called 
daylight, and gives to the sky its bluish aspect. But for this diffusive 
power of the air, no object could be visible out of direct sunshine ; the 
shadow of every passing cloud would be pitchy darkness, the stars would 
be visible all day, and every apartment into which the sun did not throw 
his direct rays would be involved in total obscurity. In ascending to 



14 



SPHERICAL ASTRONOMY. 



the summits of high mountains, the diffused light becomes less and fess, 
the sky deepens in hue, and finally, at great altitudes, approaches to total 
blackness. 

§ 71. The superior illumination of the atmosphere produced «by the 
solar light obliterates, as it were by contrast, the light from almost all 
the other heavenly bodies, and few, if any, of the latter are seen when 
the sun is up. 

REFRACTION. 

§ 72. Luminous waves which enter the atmosphere obliquely are, ac- 
cording to the laws of optics, deviated by the latter from their course, and 
made to exhibit the objects from which they proceed in positions different 
from those they actually occupj, and thus false impressions are produced 
in regard to true places of the heavenly bodies. 

Take, for example, a spectator f\g, 86. 

on the earth at A; and let LD L 
represent a section of the supe- 
rior limit of the atmosphere, and 
K A A' that of the earth's sur- 
face by a vertical plane. A star 
at S would, in the absence of 
the atmosphere, appear in the 
direction AS; but in reality, 
when the portion of the luminous 
wave moving on this line reaches 
the point i>, it is turned down- 
ward, and made to come to the 
earth at some point A\ pursuing j 

a course such as to bring its suc- 
cessive positions normal to some 

curve, as D A\ whose curvature increases towards the earth's surface, in 
consequence of the increasing density of the atmosphere in that direction. 
This part of the wave cannot therefore go to the spectator. Not so, how- 
ever, with a portion of the same general wave incident at some point as 
2>\ nearer to the zenith ; this, after pursuing a path D'A similar to D A\ 
will reach the spectator at A, and cause the body from which it originally 
proceeded to appear in the direction A S\ tangent to the curve at the 
point A, the effect being the same as though the body had shifted its 
place towards the zenith by the angular distance S A S'. 

§ 73. The air's refraction, therefore, diminishes apparently the zenith 




rf 



REFRACTION. 15 



\ 



distances of all bodies, and increases their altitudes. Any body actually in 
the horizon will appear above it, and any body apparently in the horizon 
must be below it. 

§ * 74. It is also obvious that refraction cau only take place in the ver- 
tical plane through the body, since this plane is always normal to the 
surfaces of the atmospheric strata, and divides them symmetrically. Re- 
fraction will not, therefore, in general, affect the azimuth of a body. 

§ 7o. This apparent angular displacement of a body from its true 
place, caused by the action of the atmosphere upon its luminous waves, is 
called refraction ; and various formulas have been constructed to compute 
its exact amount. One of the best of these is by Littrow, which has the 
merit of depending upon no special hypothesis in regard to the constitu- 
tion of the atmosphere, being constructed upon the most general princi- 
ples^ aud from known and well-ascertained data. 
§ 76. Make, 

Z = Z r A S' = observed zenith distance ; 
r = S A S' = corresponding refraction ; 

k = height of mercurial column, which the atmosphere supports ; 
t = temperature of the air and of the mercury ; 
a = coefficient of atmospheric expansion for each degree of Fahr.; 
/3 = coefficient of expansion for mercury, same thermometric scale. 

Then, Appendix No. III., 

r=57".82. — .]^~^-.ta.n-Z. A -0.0012517 sec 2 Z-f 0.00000139 ,'" Z \ (2) 

or, omitting the last term in the parenthesis as being insignificant for or- 
dinary zenith distances, 

r = 57".82. A. ll^Ll-M. tan Z . (1 - 0.0012517 sec 2 Z) . . (3) 

When h — 30, and t — 50, equation ( 3 ) becomes 

r m = 57".82 tan Z (1 -0.0012517 sec 2 Z) ^ A . . (4) 

and the results given by this formula for different values for Z are called 
mean refractions ; and for any other state of the thermometer and barometer, 

h 1 + (50 - 1)3 

r = A . — . — — — . 

30 1 -f- (i — 50)a' 

and taking logarithms, 

a , i h , i 1 + (50 — t)/3 

log r = log A + log h log J ... (5 ) 

b s> ^»30^ & l + ^ — 50)ct K °' 



16 SPHERICAL ASTRONOMY. 

Causing Z to vary from 0° to 90°, h from 28 to 31 inches, and t from 
80° to 20°, the logarithms above may be computed and tabulated for 
future use, under the heads Z, t, and b. 

§ 77. Causing Z to vary from 0° to 90°, in equation (4), we mav 
construct Table I.; causing t to vary from 80° to 20°, and h to vary from 
31 to 28, in the last two terms of equation (5), we may construct Table 
II. Returning to equation (2), resuming the quantity omitted to obtain 
equation (3), computing their values for zenith distances, varying from 
7o D to 90°, on the supposition that A = 30 and £=50, an additional table 
may be computed to correct the refractions in low altitudes. Tables L, 
11., and III. are due to Mr. Ivory. 

§ 78. For zenith distances exceeding 80°, refraction becomes very 
uncertain ; it then no longer depends solely upon the state of the atmo- 
sphere, which is indicated by the barometer and thermometer, being fre- 
quently found to vary at the same, station some 3 to 4 minutes for the 
same indications of these instruments. 

Example. — The zenith distance of an object is observed to be 71° 26' 00", 
the barometer standing at 29.76 in., and the thermometer at 43° Fahr. : 
required the refraction. 

Table I. Mean refraction, log. 2.23609 

Table II. Barometer 29.76 " 9.99651 

Table II. Thermometer 43° " 0.00668 



Refraction 2' 53".49 . . 2.23928 

Observed zenith distance . . . 7l° 26' 00".00 



Zenith dist. cleared from refraction 71° 28' 53 ".49 



The refraction must always be added to the observed zenith distance, or 
subtracted from the observed altitude, to clear an observation from re- 
fraction. 

PARALLELISM OP THE EARTH'S AXIS, AND UNIFORMITY OF THE 
EARTH'S DIURNAL MOTION. 

§ 79. Wherever upon the earth's surface the altitudes and instru- 
mental azimuths of a star are taken in the various points of its diurnal 
course, and the instrument is turned in azimuth, so as to read the half sum 
of two azimuths, corresponding to any two equal altitudes, the vertical 
plane through the line of collimation is found to divide the path symmet- 



PARALLELISM OF THE EARTH'S AXIS. 17 

rically ; and this plane of symmetry for any one star will, at the same 
place of observation, also be a plane of symmetry for all the stars. In 
other words, the diurnal paths of the stars may be divided symmetrically 
by any number of planes inclined to one another through the earth's 
centre — a condition which can only be fulfilled for paths upon the celes- 
tial sphere, when these paths are circles, of which the poles coincide, and 
the planes of symmetry pass through them. 

The diurnal motions of the stars are only apparent, and arise from an 
actual motion of the spectator about the earth's axis. This latter line 
preserves, therefore, its direction unchanged, and, in the motion of the 
earth around the sun, describes a cylindrical surface, of which the elements 
have their vanishing point in the poles of the celestial sphere. These 
poles are therefore the geometric poles of the diurnal paths of the stars, 
and the planes of symmetry are the meridian planes of the places of 
observation. 

§ 80. Again, the interval of time during which a star is moving be 
tween any two given altitudes on one side of the plane of symmetry, is 
exactly equal to that during which it is moving between the equal alti- 
tudes on the opposite side, which can only be true, for all positions of the 
observer, when the star's apparent, or the earth's real motion about its 
axis, is uniform. 

§ 81. The period of one revolution of the earth about its axis is called 
a day; the day is divided into 24 equal parts called hours; the hours 
into 60 equal parts called minutes ; the minutes into 60 equal parts called 
seconds, and die seconds into 60 equal parts called thirds. 

§ 82. The earth rotates therefore at the rate of 360-^24=15° an 
hour; 15' of space in 1 minute of time; 15" of space in 1 second of time, 
or 15'" of space in 1 third of time. 

§ 83, Distances on the equinoctial may therefore be expressed in 
time or space at pleasure, the former being convertible into the latter by 
multiplying by 15, or the latter into the former by dividing by 15. 

§ 84. To distinguish hours, minutes, and seconds in time, from degrees, 
minutes, and seconds in arc, the former are usually designated by the nola 
tion h, m, s, and the latter by °, ', " ; thus an arc upon the equinoctial 
may be written 357° 39' 38", or 23 h 50 m 38 s .5. 

§ 85. To find the instrumental azimuth of the meridian of a, place. — 
Bring the line of collimation of an altitude and azimuth instrument, prop- 
erly levelled, upon a star in the east or west, clamp the vertical circle, and 
read the instrumental azimuth; then by an azimuthal motion bring the line 
oi collimation upon the star when in the west or east, and again read the 

2 



18 



SPHERICAL ASTRONOMY 



azimuth : the half sum of the two will be the instrumental azimuth sought. 
To bring the line of collimation into the meridian, turn the instrument till 
it reads this half sum. 






UPPER AND LOWER DIURNAL ARCS.— CIRCUMPOLAR BODIES. 

§ 36. The diurnal paths of the heavenly bodies which are cut by the 
horizon are, in general, divided by the 
latter unequally. The portions of these 
paths above the horizon are called the 
upper, and those below the lower diurnal 
arcs. 

§ 8*7. To find, for any spectator, thb 
relation which these arcs bear to one an- 
other, let PQP' Q' be the meridian, P 
the elevated pole, Q Q' the equinoctial, 
Z the zenith, H W H' the horizon, 
S'SS"S'" the diurnal path of any 
body, the earth being a mere point at E\ 
then will S' $S" be the upper, and S" S r " S' the lower diurnal arc. 



I = QZ, latitude of the spectator, 
p = P S", polar distance of the body, 

P = ZP S'\ the hour angle of the body when in the horizon, 
z = Z S", zenith distance of the body in horizon. 

I 




Then in the triangle ZP S", because PZ = 90 

cos z = cos p sin I + sin p cos I cos P 
but z = 90°, whence 

= cos p sin I + sin p cos I cos P ; 



or 



cos P = — 



tan I 
tan p 



(6) 



m 



l£lz=z0,OTp = 90°, then will 

cos P = 0, and P = 90° == 6 h ; 

that is, if the spectator be upon the equator, or the body upon the equi- 
noctial/the semi-upper arc will be six hours, and the body will be as long 
above as below the horizon. 



'r ^ 






CI HC UM POLAR BODIES. 



P \<?D 



-.-'-< 



if p < I, then will 

cos P < — 1 ; 

which is impossible, and the place of the body can never satisfy the con- 
dition that z = QQ°. In other words, when the [olar distance is less 
than the latitude of the spectator's place, the body can never sink to the 
horizon, and will ever remain in the field of perpetual apparition. Such 
bodies, as well as their diurnal paths, are said to be circumpolar. 
If p = /, then will 

cos P = - 1 ; P = 180° = 12 h ; 

that is, when the polar dis ance of the body is equal to the la*i f ude of 
the spectator's place, the body can never sink below the horizon, but will 
just graze it in the meridian. 
If>> I, andp <90°, 

cos P < 0, cos P > - 1 ; P > 90°, P > 6 h ; 

that is, all bodies between the eleva'ed pole and the equinoctial, will be 
longer above than below the horizon. 
If p > /, and p > 90°, 

cos P > 0, cos P < 1, P < 90°, P < o h ; 

that is, if the body and the spectator be on opposite sides of the plane of 
the equinoctial, the semi-upper arc will be less than six hours, and the 
body will be a shorter time above than below the horizon. 
If p — 180° — /, then will tan p = — tan /, and 

cosP= 1, P=0° = h ; 

that is, when the body is at a distance from the depressed pole equal to 
the latitude of the place, the body will never rise above the horizon, but 
just graze it in the meridian. 

If p > 180° — I, then will tan p > - tan L, and 

cos P > 1, 

which is impossible. That is to say, if the body's distance from the de- 
pressed pole be less than the spectator's latitude, the body can never ris 
to the horizon, and must ever remain invisible. 

§ 88. The act of a body's passing the meridian, is called its cttlmina 
tion. A body has its greatest or least altitude at the instant of its cui 
mination. The altitude of a body when on the meridian is called its 
meridian altitude. 



20 



SPHERICAL ASTRONOMY. 



TERRESTRIAL LATITUDE AND LONGITUDE 



§ 89. Latitude.— When in Eq. ( 1 ) the angle Z P S" =P=180 W , 
then will p = I ; but in this case p is the polar distance of the point of the 
horizon of the same name as the elevated pole, and hence the latitude of the 
spectator is always equal to the altitude 
of the elevated pole. 

§ 90. This suggests an easy and 
accurate method of getting from obser- 
vation both the latitude of the specta- 
tor's place and the polar distance of a 
star. 

Let Z be the zenith, HH' the hori- 
zon, Q Q' the equinoctial, P the eleva- 
ted and P r the depressed pole, and S r S, 
the diurnal path of a circumpolar star. 
Make 

I = HP = Z Q , the latitude, 
p = P S r = P S , the polar distance of star, 
a' = US' , the greatest observed meridian altitude of star, 

a { = US , the least observed meridian altitude of star, 

r' and r t 7 the refractions corresponding to the greatest and 




Then from the figure will 

V + a, 



1 = 



least meridian altitudes respectively. 



__^+a / -(r r +> / ) 



(8) 



p = 



(a-r) a '-a-(r'-r) 



(9) 



That is to say, the latitude of the observer's place is equal to the half sum 
of the greatest and least meridian altitudes of a circumpolar star ; and the 
polar distance of the star is equal to the half difference of its greatest and 
least meridian altitudes. Other methods for finding the latitude will be 
given in another place. 

§ 91. Longitude. — The uniform motion of the earth about its axis fur- 
nishes the means of finding the longitude of the spectator's place. 

Twenty-four perfect time-keepers, with dial-plates graduated to 24 hours, 
placed upon meridians 15° apart, and so regulated as to mark 24^ at the 
instant any one fixed star or other point of the heavens culminates, would, 



/ 




FIGURE OF THE EARTH. 21 

§ 82, when this regulating star or point comes to any one of these me- 
ridians, simultaneously mark the hours indicated by the natural numbers 
from one to twenty-four, inclusive; that 15° to the east of the regulating 
point marking l h , that 30° to the east marking 2 h , and so on to that 345° 
o the east, or 15° to the west, marking 23 h . The timepieces to the east 
would be later and later, those to the west earlier and earlier. The times 
indicated on these several timepieces are called the local times of their re- 
spective meridians. 

§ 92. If now, without altering its hands or rate of motion, a traveller 
were to transport the time-keeper of any one of these meiidians to that on 
any other, and note the difference of time indicated by the two, this differ- 
ence would be the difference of longitude of the two meiidians, expressed 
in time; and multiplied by 15 would give the same in degrees. 

§ 93. If one of these meridians be the first meridian, this difference 
would be the longitude of the other. But if neither be the first meridian, 
this difference applied to the longitude of one, supposed known, would give 
the longitude of the other. 

§ 94. The solution of the problem of longitude consists, therefore, in 
finding the difference of the local times which exist simultaneously on the 
first and required meridians. The various modes of 'doing this will be 
given in another place. 



FIGURE AND DIMENSIONS OF THE EARTH. 

§ 95. A fluid mass rotating about an axis, and of which the particles 
attract one another with intensities varying inversely as the square of their 
distances apart, will assume the form of an oblate spheroid. Its axis of 
rotation will be both the shortest and a principal axis of figure. Where 
the angular velocity is such as to make the centrifugal force of the sur- 
face elements small in comparison with their weight, due to the attraction 
of the whole mass, the figure of the meridian section will., (§ 265, Analyt. 
Mechanics]) approach that of an ellipse of small eccentricity. 

§ 96. The centrifugal force of a body at the equator of the earth, 
where it is greatest, is only about ¥ |-^th part of its weight. Observations 
upon the temperature of the strata composing the earth's crust, lead to 
the conclusion that at no great depth below its surface its materials are in 
a fluid state from excessive heat; and the researches of geol )gy make it 
inore than probable that there was a time when the earth was without 
solid matter. Its present irregularities of surface, forming mountains, 
hills, valleys, the bed of the ocean, of seas, lakes and rivers, are due to 



22 



SPHERICAL ASTRONOMY, 



changes subsequent to the surface induration from cooling, and as the ver- 
tical dimensions of these a:e insignificant in comparison with the depth 
to the centre of the entire mass, it is concluded that the figure of the 
earth is one of fluid equilibrium due to its rotary motion. 

§ 97. Assuming the meridian section of the earth to be an ellipse, its: 
eccentricity and semi-axes are found, Appendix No. IV^ from the relations 



P * = a 



A = 



B = A.Vl 



csm 1 l M — c' siu*/ r ^ 
■ t (l— ^sin*/,)' 



(10) 
(12) 



r 



in which 

e = the eccentricity of the meridian ; 
A = semi-transverse axis = equatorial radius of the earth ; 
B — semi- conjugate axis = polar radius of the earth ; 
€ and c' = the linear dimensions of the ares of the meridian, whose 
extremities differ in latitude by 1 ° ; 
l 7 m and l' m = latitudes of the middle points of the arcs e and e' respec- 
tively. 

The quantities /,„, V m , c, c r are found from observation and measure- 
ent. A method by which / and /' may be found is explained in § 90. 

§ 98. To find € and c, a base line AB is carefully measured on some 
extended plain, and a number of stations C, 1), E, F, H, &t\ are se 
Ifccted in a northerly or southerly diiection, and so that C 
may be seen from A and B, D from B and C, E from C 
and I), and so on to the end. The several stations being 
connected by right lines, a network of triangles is formed ; 
every angle in each triangle is carefully measured, and the 
instrumental azimuth of its vertex, and that of the meridian, 
as viewed from the other two t accurately noted. (§ 85). 
The angles being cleared from spherical excess, the sides of 
the triangles are then computed, beginning of course with 
the triangle of which the measured base is one of the sides. 
The difference between the instrumental azimuths of the 
several vertices and those of the meridian, gives the inclina- 
tion of the sides to the meridian line. The product of each 
side into the cosine of its inclination gives the projection of this side on 
the meridian, and the sum of the projections $k&®f^&m of the series of 
sides, as AB, BC, CD, I J E, EH, and II F, connecting the most north- 




\ 



FIGURE OF THE EARTH 23 

erly and southerly points, will give the linear meridional distance L L\ 

between the parallels of .atitude through the same points. 

Make 

a = the sum of these projections, expressed in miles ; 

l K — the latitude of A, supposed the most northerly ; 

l s = the latitude of F, supposed the most southerly ; 

then, 

l a -l. : 1° : : a : c, 

whence t 

a 

and 

m ~ 2 * 

The same operations being repeated in a different locality considerably 
further north or south, the values of c' and l' m are found, and hence from 
equations (10), (11), and (12), the dimensions of the earth. 

From the arcs known as the Peruvian, Indian, French, English, Hano- 
verian, Danish, Prussian, Russian, and Swedish, names derived from the 
countries in which the arcs were mostly measured, Bessel found, 

e 2 = 0.0068468, 
2 A = 7925,004 miles, j 
2B = 7899,114 miles, V ...... . . (13) 

Polar compression . = 26,490 miles. ) 

§ 99. By the ellipticity of the earth is meant the difference between 
its equatorial and polar radii, expressed in terms of the equatorial radius 
as unity. Denoting the ellipticity by E, we have 

S = —j- = TO nearly (14) 

§ 100. The length of a degree of latitude, denoted by /3, in any lati- 
tude /, is, Appendix No. IV, equation (/), given by 

2 * A 1 - e* 

• 1bU (1 _ ,2 sin 2 ^2 

The length of a degree, measured perpendicularly to the meridian, de- 
noted by /? h is, Appendix No. IV., equation (ra), given by 

%*■ ' J f^ e 2 sin 2 l , , 

^rW'^^i^-pi • • • • ( 16 > 



21 SPHERICAL ASTRONOMY. 

and the length of a degree of longitude, denoted by a, measured on a par- 
allel of latitude iu the latitude Z, is, App. No. IV., equation (o), given by 

2 if cos / 

'-sao-^ynrm (17) 

§ 101. The close agreement between the results of these formulas and 
those of actual measurement, at various and numerous places on the earth, 
justifies in the fullest manner the assumption in regard to its ellipsoidal 
figure. 

The equatorial circumference of the earth is 24,899, say, for convenience 
of memory, 25,000 miles. The lengths of the degrees of latitude increase 
from the equator to the poles. In the latitude of 50° the length is about 
70 statute miles, and contains nearly as many thousand feet as the year 
contains days (365), and each second is equivalent to about 100 feet. 



GEOCENTRIC PARALLAX. 

§ 102. The bodies of the solar system being comparatively near to the 
earth, a change in a spectator's place on the earth's surface gives to them 
a sensible parallactic motion on the surface of the celestial sphere, and 
two observers at remote stations would not assign to these bodies the same 
places at the same time without first clearing their observed co-ordinates 
of this source of discrepancy. The mode of correction is to refer all obser- 
vations to one common station, and this station is assumed, for conve- 
nience, to be at the centre of the earth. 

§ 103. The place in which a body would appear, if viewed from the 
centre of the earth, is called its Geocentric Place. 

§ 104. The apparent change of a body's place that would arise from a 
change of the spectator's station from the surface to the centre of the 
earth is called Geocentric Parallax. . 

§ 105. The transfer of station from the surface to the centre of the 
earth is sensibly in a vertical circle, and the geocentric parallax is there- 
fore in the same plane. 

§ 106. The co-ordinates of a body's place, as determined by observa- 
tion, corrected for geocentric parallax, are the geocentric co-ordinates of 
the body. 

§ 107. The point in which the radius of the earth produced through 
the spectator's place pierces the celestial sphere, is called the central zenith. 
The arc of the celestial meridian from the central zenith to the equinoo- 



GEOCENTRIC PARALLAX. 



25 



Fig. 40. 



tial, is called the central latitude. The difference between the latitude 
and the central latitude, is called the reduction of latitude. 

Thus BAB' A', being- a meridian 
section of the terrestrial spheroid, and 
Z Q an arc of the celestial sphere in the 
same plane, M the spectator's place, Q 
the highest point of the equinoctial, 
MO the direction of the plumb-line, 
CM the radius of the earth ; then will 
Z' be the central zenith, Z' Q the cen- 
tral latitude, and ZMZ' = Z Q—Z'Q, 
the reduction of latitude. 

§ 108. Denote in future the central 
latitude by £, the polar radius by y, and the latitude by /', then, Appen- 
dix No. IV., equation (</), 




tan I = y 2 tan /' 



(18) 



that is, the tangent of the central latitude is equal to the tangent of the 
latitude into the square of the polar radius. 

Denote the radius of the earth drawn to the spectator's place by p, 
then, Appendix No. IV., equatiou (r), 

1 

■ f .' . . . . (19) 



7 

Thus, the latitude being found from observation (§ 90), the centra* 
latitude becomes known from equation (18), and hence the radius of the 
earth drawn to the spectator's place, equation (19). 

§ 109. Let AB'A'B be a meridian «*«. 

sectiou of the earth's surface, A A' the 
equatorial diameter, M x and M 2 the 
places of two observers viewing the same 
body S. The observer at M x would see 
the body projected upon the celestial 
sphere at S h that at M 2 would see it 
projected at S Si and to an observer at 
the centre it would appear at S 3 . The 
points Z, and Z 2 are the central zeniths 
of the two observers ; Z, Si and Z 2 £ 2 

are the central zenith distances of S, as viewed from Mi and J/ 2 respec- 
tively. The first diminished by S s 5, and the second by £ 3 $. 2 , will give 




26 



SPHERICAL ASTRONOMY. 




When the body is 



Z x S 3 and Z 2 S 3 the central zenith dis- Fig. 41 bis. 

tances as they would appear from the 

centre. Bat S 3 S { measures the angle 

S 3 S £, = M, S C, and S 2 S- s the angle 

S 2 S S 3 = M 2 S C ; so that the angles 

M £(7 and M 2 S C are the corrections 

for parallax. 

'§ 110. The parallax of a body is the 
angle at the body subtended by the 
earth's radius drawn to the spectator. 

When the body is above the horizon, 
or is in altitude, it is called the parallax in altitude. 
in the horizon, it is called the horizontal parallax. 

Make 

2, = M x S C = parallax in altitude at M x ; 
z 2 ^= M 2 S C = parallax in altitude at M 2 ; 
P, = horizontal parallax at M x ; 
P 2 = horizontal parallax at M 2 ; 

= distance of the body from the earth's centre ; 
= radius of the earth for M x ; 
= radius of the earth for M 2 ; 
= central zenith distance at M x ; 
== central zenith distance at M 2 : 
l x — Z X C A — central latitude of M x ; 
l 2 — Z 2 C A = central latitude of M 2 . 



r= CS 
Pl = Ii x 

P2 



z x = z x s x 
z 2 = z 2 s 2 



Then, in the triangle M x S C, 



whence 



sin z x : sin Z x : : p t : r ; 

sin z x = — . sin Zj ; 



But because z x is always very small, we may write 



in which u is the number of seconds in radius, and 2j is expressed in the 
same unit ; which substituted above gives 



2, = w . — . sin Zi 
r 






GEOCENTRIC PARALLAX. 27 

when Z, becomes 90°, the body is in the horizon, and 2, becomes P,, and 
we have 

n=»-| (20) 

and this above gives 

z,= P,.sin Z, . (21) 

Whence the parallax in altitude is equal to the horizontal parallax into 
the sine of the central zenith distance. 

§ 111. If the observer be upon the equator, then will p, become unity, 
P, becomes the horizontal parallax on the equator, called the equatorial 
horizontal parallax ; designating this latter by P, we have, equation (20), 







P = 


r 


and this in 


equation (20) 


gives 








Pi = 


:P. 



P. ( 22 ) 

that is to say, the horizontal parallax of a body at any place, is equal to 
the product of the equatorial horizontal parallax of the body by the ra- 
dius of the earth at the place. 

The value of P, in equation (21) gives 

z l = P. ?l .smZ l ....... (23) 

§ 112. To find the equatorial horizontal parallax of any body, we 
havu in the triangles M x S C and M^S C 

z l = P. ?i .smZ 1 

z 2 = P.p 3 .siuZ a 
acid ng 

z l + z 2 = P.{ h . sin Z x + Pa . sin Z 2 ), 
but 

z.^Z.-Z.CS, 

by addition 

z x + z. 2 = z, -f z 2 - (z, cs + z 2 c s) = z, + z 2 - (i x + y , 

which substituting above, and dividing by the coefficient of P, gives 

j^* + *=ft+a. w 

p, sin Z, -f p, sin Z 2 v ' 

§ 113. If the body be so remote that the difference between the radii 
of the earth, as viewed from it, be insignificant, which is the sase with all 



28 SPHERICAL ASTRONOMY. 

bodies except the moon, p, and p 2 may be regarded as equal to one an- 
other, and each equal to unity, and we shall have, equation (24), 



sin Z x + sin Z 2 



(25) 



in which /, and l 2 are the central latitudes of the places M x and M % , 

§ 114. In all this the observers have been supposed to be on the same 
meridian ; but this is not necessary, nor would it, in general, be the case 
in practice. If on different meridians, make 

8 = change of meridian zenith distance of the body in the interval 

between two consecutive culminations ; 
X = difference of longitude of the two observers, expressed in time ; 
8 / =z change in meridian zenith distance while passing from the first 
to the second meridian ; 
then 

24 h : 8 : : X : 8' ; 
whence 

s '=^i ■• • <*> 

If the meridian zenith distance be increasing at the easterly station, 8' 
is to be added to, if decreasing, subtracted from, the meridian zenith dis- 
tance at that station. This corrected meridian zenith distance will be 
that which the body would have to an observer on the meridian of the 
westerly station, and on the same parallel of latitude with the observer 
on the easterly meridian, the reduction being in effect to bring the ob- 
servers to the same meridian. 

§ 115. To recapitulate: the latitudes of two stations are first found 
from observation ; the central latitudes are found from equation (18) ; the 
radii of the earth at the two stations, fiom equation (19) ; the equatorial 
horizontal parallax, from equation (24) ; the horizontal parallax at any 
place, from equation (22) ; and the parallax in altitude, from equation (23), 

AUGMENTED AND HORIZONTAL DIAMETERS. 

§ 116. By the rotation of the earth upon its axis the spectator is con- 
tinually changing his distance from the heavenly bodies. A change of dis- 
tance gives rise to a change in the apparent dimensions of an object. A 
body seen in the horizon of a spectator would appear to him sensibly of 
the same size as if seen from the centre of the earth, the distances IT C 
and H ' M y for the nearest of the heavenly bodies, being sensibly the same. 



DIMENSIONS OF THE HEAVENLY BODIES. 



29 





The apparent semi-diameter of a body 
is the angle at the observer subtended by 
the body's real serai-diameter, the latter 
being perpendicular to a visual ray drawn 
to one of its extemities. 
Make 
= HMB = apparent semi-diameter of 

a body when in the horizon ; 
= LMB / = apparent semi-diameter of 

f£*e borfgjrhen in altitude ; 
= HE = Jj^H= real semi-diameter of the body in linear units; 
r =: body's «^HTce from the observer when in the horizon = distance 
^^mm earth's centre ; 
ody's distance from the spectator when in altitude; 
Z MwF=z the body's central zenith distance; 
% = M L C ' = the body's parallax in altitude. 

Then 

r . sin * = d = r 7 sin s / ; 
whence 




r sin Z 



r 



ry 



r sin (Z—z) cos Z 

iMtM +* \a *J COS Z : — 



sinz 



6 

repladfcg sin z by its value — , and z by its value in Eq. (23), also writing 



rsir 



for sin s, and - for sin s f , we find 



COS z 



p 7 

Pi' cos Z 

' x /.I 



. . . . . (27 ) 



in which s' and s are expressed in seconds of arc. 

§ 117. The apparent diameter 2 s of a body in the horizon, is called the 
horizontal diameter ; its apparent diameter 2 s f in altitude, is callea the 
augmented diameter. 

DISTANCES AND DIMENSIONS OF THE HEAYENLY BODIES. 

§ 118. Having found the horizontal parallax of a body, it is easy to find 
its distance from the earth's centre. From equation (20) we have 






(28) 



P 
in which p and P are respectively the equatorial radius of the earth and 



m 




30 SPHERICAL ASTRONOMY. 

the equatorial horizontal parallax of the body ; and from which we con- 
clude that the distance of any body from the earth's centre, is equal to the 
equatorial radius of the earth repeated as many times as the number of 
seconds in the body's equatorial horizontal parallax is contained in the 
number of seconds in radius. 

§ 119. The horizontal parallax of a body is the apparent semi-diameter 
of the earth as seen from the body. The apparent semi-diameter of t 
bodies seen at the same distance are directly proportional to their rea 
magnitudes. Make 

s = apparent semi-dhmeter of the body ; 

d — the real semi-diameter of the body in litjjjjhunits, as miles i 

P = the eqiatorial horizontal parallax of thel 

p = the equatorial radius of the earth ; 

then will 

P : s : : p : c?; 
whence 

rf=P-j>- (29) 

that is, the real semi-diameter of any heavenly body is equal to the equa- 
torial radius of the earth repeated as many times as the body's equatoiial 
horizontal parallax is contained in its apparent semi-diameter. 'fjThe appa- 
rent diameter of a body is measured by means of the micrometer. 



ECLIPTIC. 

§ 120. The orbit of the earth about the sun is sensibly a plane curve. 
The intersection of the plane of the earth's orbit with the celestial sphere 
is called the ecliptic. The ecliptic is a g*eat circle of the celestial sphere 
because its plane passes through the earth's centre. 

§ 121. The orbital motion of the earth about the sun gives rise to a 
parallactic motion of the sun about the eai th, and the effect to a spectator 
on the earth is the same as though the latter were stationary and the sun 
in motion about the earth. The sun appears to move along the ecliptic in 
the same direction that the earth's projection upon the celestial sphere, as 
seen from the sun, actually moves in that great circle. 

§ 122. The earth's axis being oblique to the plane of the ecliptic, forms 
an angle with the radius vector of the earth. The axis of the earth re- 
taining its direction sensibly unchanged, this angle is variable. 

§ 123. Let S be the sun, E I E II E II) E IIII the earth's orbit, PS a line 
through the sun's centre and oarallel to the direction of the earth's axis, 






ECLIPTIC, 



31 




E l E tll the projection of this line on the plane of the ecliptic. Draw 
E tl & E ,ur perpendicular to E t E ul , and E t P ( , E lt P u% E ul P til , and 
E IUi P llu parallel to SP. The angle P,E t S will be the polar distance 
of the sun when the earth is at E t , P ' it E n S when at E u , P UI E UI S 
when at E UI , and P nil E Ujl S when at E llu . It will be least at E^ 
greatest at E tti , and 90° at E n and E nu . The polar distance at E //f is 
the supplement of that at E n and estimated from the nearest or opposite 
poles, the polar distances at E / and E u/ are equal. 

§ 124. The declination being the complement of the polar distance, the 
eun's declination will sometimes be north, sometimes south. Its north de- 
clination will be greatest when its north polar distance is least ; its south 
declination greatest when its south polar distance is least. 

§ 125. Thus, by the orbital motion of the earth, the terrestrial equator 
is carried from one side of the sun to the opposite, and the sun itself made 
apparently to pass alternately from north to south and from south to north 
of the equinoctial. 

§ 126. The radius vector would, by the orbital motion alone, trace upon 
the surface of the earth an ellipse of which the plane would coincide with 
that' of the ecliptic; by the diurnal motion alone, it would trace a parallel 
of latitude ; and by both motions combined, it actually describes a kind of 
spiral curve extending on either side of the equator and intersecting all the 
parallels between those whose latitudes are equal to the sun's greatest 
north and south declinations. To spectators situated somewhere on these 



32 



SPHERICAL ASTRONOMY. 




parallels, the sun will be vertical, or in the zenith, twice in the course of 
one revolution of the earth about the sun. 

§ 127. Two small circles of the celestial sphere parallel to the equinoc- 
tial and at a distance therefrom equal to the sun's greatest north and south 
declination are called tropics ; that on the north is called the tropic of 
Cancer, and that on the south the tropic of Capicorn. 

§ 128. A plane through the earth's centre, and perpendicular to the 
radius vector, divides the earth's surface into two equal parts. That on 
the side towards the sun is illuminated, while that on the opposite side is 
in the dark. To an observer on the former it is day ; to one on the latter 
it is night. The curve which separates the enlightened portion from the 
dark is called the circle of illumination. 

§ 129. When the earth is in either of the positions E,, or E ///n its 
axis is in the plane of the circle of illumination ; this latter divides all par- 
allels equally, and the lengths of the days are equal to those of the nights 
all over the earth's surface. 

When the earth is in either of the positions E, or E //n its axis makes 
the greatest angle possible with the plane of the circle of illumination ; the 
latter divides the parallels most unequally, and the length of the days will 
differ from those of the nights the most possible. 

§ 130. To all places north of the parallel whose latitude is equal to the 
north polar distance of the sun, the sun will, Eq. (7), be circumpolar, 
while to all places having an equal south latitude the sun will not, Eq. (7), 



ECLIPTIC. 33 

rise. In like manner, to all places south of the parallel whose latitude is 
equal to the south polar distance of the sun, the sun will be circuuipolar ; 
while to all places of equal north latitude, the sun will not rise. 

§ 131. During the time the earth is moving from E„ to E,„„ the sun 
will shine upon the south pole, and the north pole will be deprived of his 
direct light. While moving from E ////t to E„ the reverse will be true. 
The zones of polar illumination and obscuration will increase from E„ 
to E /// and from E„„ to E„ and diminish from E, to E„ and from E y/ , 
to E ///r The radii of the greatest zones of polar illumination and obscu- 
ration for one diurnal revolution are P,I, and P,„I„„ which are equal to 
one another, being the greatest departure of the pole from the circle of il- 
lumination on opposite sides. 

§ 132. Draw E, Q, and E„, Q,„ respectively perpendicular to P,E, 
and P^E,,,; then will Q' Q t and Q"' Q„, represent the equator, Q 4 D t 
and Q,„D,„ the greatest north and south declinations of sun respectively, 
and P, I, and P U4 I Ui the greatest departure of the pole from the circle of 
illumination. Now 

I, D, = P,<Z, = 90° = /„, D lu = P UI Q„„ 

and by subtraction 

/',/../<.',>, P.J.^Q.uD,.,; 

that is, the radius of the zone of greatest polar illumination, or obscuration, 
is equal to the greatest declination of the sun. /~ > 

§ 133. Two small circles parallel to the e$mft&d£d, and at a distance 
from the poles equal to the greatest declination of the sun, are called polar 
circles ; that about the north pole is called the arctic, and that about the 
south the antarctic circle. The polar circles are the boundaries of the 
greatest zones of polar diurnal illumination and obscuration. 

§ 134. When the intervals of time between three consecutive passages 
of a circumpolar star over the line of collimatiou of a transit or mural cir- 
cle are equal, these instruments are adjusted to the meridian. 

§ 135. The diurnal motion brings the meridian of a place, in the course 
of one revolution of the earth on its axis, into coincidence with the decli- 
nation circle of every body in the heavens. The difference of times between 
the meridian's passing the centres of any two bodies, is the difference of 
light ascension of these bodies. 

§ 136. To find the time of the meridian's passing the centre of any 
body, find by the transit instrument and timepiece the time of tru- merid- 
ian's passing the body's east and west limb, and take half the sum. 

3 



34 



SPHERICAL ASTRONOMY. 



Fig. 44. 



§ 137. To find the polar distance of a body's centre, take the reading of 
the mural circle when its line of colli mation is upon the upper or lower 
limb ; subtract from this the polar reading and correct the difference for 
refraction, parallax in altitude, and semi-diameter. The declination is ob- 
tained by subtracting the polar distance from 90°. 

§ 138. The points in which the equinoctial intersects the ecliptic are 
called the equinoxes ; that by which the sun passes from the south to the 
north of the equinoctial is called the vernal equinox ; the other, or that by 
which the sun passes from the north to the south of the equinoctial, is called 
the autumnal equinox. 

§ 139. The angle which the equinoctial makes with the ecliptic is called 
the obliquity of the ecliptic. 

§ 140. To find the place of the vernal equi- 
nox and the obliquity of the ecliptic, let VD 2 
be an arc of the equinoctial, V S 2 of the eclip- 
tic, V the vernal equinox, aS^ and S 2 two 
places of the sun when on the meridian at 
different times, SiD u S 2 D 2 arcs of declina- 
tion circles ; and make 

S v = D x S u the sun's declination at any meridian passage ; 
62 = D 2 S 2 , the same at some subsequent passage ; 
2a = VD 2 — VDi, the corresponding difference of right ascension ; 
x = VD h the right ascension of the sun at the time of first meridian 

passage ; 
w = S x VD U the obliquity of the ecliptic. 

Then in the triangles S { VD { and S 2 VD 2 , right-angled at D x and D 2 , 

sin x = tan S x , cot w, 
sin (x + 2a) = tan § 2 . cot w ; 
and by division 

sin {x -J- 2a) tan S 2 




"Sa D, 



sin x 



tan <5, 



(30) 



adding unity and clearing the fraction, then subtracting unity ai d clear 
ing, and dividing one result by the other, we find 

sin (x + 2a) -f- sin x tan S 2 + tan <J, 
sin {x -f- 2a) — sin x ~~ tan S s — tan ^ ' 

tan (x + a) sin (8 2 -f- § x ) 
tan a "~ sin (8 a — 8 X )' 




ECLIPTIC. -35 

Whence 

tan(x + a)= ™ n( ?: + § :] .tana . . . . (31) 

• sin (o, — d,) 

Also 

cot oj = sin a; . cot <5, (32) 

The value of the obliquity is thus found to be nearly 23° 2*7' 54", which 
is therefore the greatest north and south declination of the sun. The tropics 
are, therefore, 23° 27' 54 // from the equinoctial, and the polar circles are 
at the same distance from the poles. 

§ 141. The interval of time between the sun and a star crossing tha 
meridian, applied to the right ascension of the sun, gives the right ascen- 
sion of the star. The declination of a star is found like that of the sun, 
except that there is no correction for parallax and semi-diameter, the only 
correction being for refraction. 

142. The right ascension and declination of one star being known, 
the differences of observed light ascensions and declinations, the latter being 
corrected for differences of refractions, give, when applied to the right as- 
cension and declination of the known star, the light ascension and decli- 
nation of other stars. Thus a list of the stars, together with their right 
ascensions and declinations, and arranged in the order of their right ascen- 
sions, furnishes the ground-work of what is called a catalogue of stars, of 
which a fuller account will be given presently. 

§ 143. A belt of the heavens extending on either side of the ecliptic, 
far enough to embrace the paths of the planets, is called the zodiac. 

§ 144. The ecliptic is divided into twelve equal parts, called signs. 
They commence at the vernal equinox, and are named in order, proceed- 
ing towards the east, Aries (t), Taurus (tf), Gemini (n), Cancer (-3), 
Leo ($1), Virgo (HjL), Libra (=^=), Scorpio (%), Sagittarius ( 1 ), Caprkor- 
nus (V?), Aquarius ( ~), and Pisces (X). Motion in the order of the 
signs is said to be direct ; the converse, retrograde. 

§ 145. The points of the ecliptic in which the sun reaches his greatest 
north and south declination are called the solstitial points : that on the 
north is called the summer solstice, and that on t\\e, south the whiter sol- 
stice. The sun when in these points appears to be stationary as regards his 
apparent motion in declination. The solstitial colure is the declination 
circle through the solstitial points. The equinoctial colure is the declina- 
tion circle through the equinoctial points. The solstitial colure separates 
Gemini from Cancer, and S.igittarius from Oapncornus; the equinoctial 
colure separates Aries from Pisces, and Virgo from Libra. 

§ 14G. A great circle of the celestial sphere passing through the poles* 
of the ecliptic is called a circle of latitude. 



SPHERICAL ASTRONOMY. 




§ 147. The latitude of a body is the distance of the booy's ,entre from 
the ecliptic, measured on a circle of latitude. 

§ 148. The longitude of a body is the distance from the vernal equinox 
to the circle of latitude through the body's centre, measured on the eclip- 
tic in the order of the signs. 

The longitude and latitude are co-ordinates that refer a body's place to 
the circle of latitude through the vernal equinox and to the ecliptic ; the 
longitude and ecliptic polar distance are polar co-ordinates that refer a 
body's place to the same circle of latitude and to the pole of the ecliptic- 

§ 149. The longitude of the sun, as seen from the earth, is readily ob- 
tained from the obliquity of the ecliptic and either the right ascension of 
declination. 

For this purpose make Yi s- u bis - 

a = VSi, the longitude of the sun ; 

8 = Si D u his declination ; 

a = VD U his right ascension ; 

w •== S { VD i7 the obliquity of the ecliptic. 

Then will 

tan a 




tan a 



cos w 



(33) 



sin w 



(34) 




§ 150. The place of the sun as seen from the earth, and that of the 
earth as seen from the sun, are at the opposite extremities of the same di- 
ameter of the ecliptic; and the longitude of the sun, increased by 180° f 
will be the longitude of the earth as viewed from the sun, the centre of the 
earth's orbital motion. 

§ 151. The sun appears in the vernal equinox on the 20th March, in 
the autumnal equinox on the 22d September, the summer solstice on the 
21st June, and in the winter solstice on the 21st December. 

The poles of the ecliptic are at a distance from the nearest poles of the 
^equinoctial, equal to the obliquity of the ecliptic. 

/ § 152. The right ascension is obtained from observation by means of 

/ the clock and transit instrument, the declination by means of the mural 

circle. From these and the obliquity of the ecliptic, the longitude am J 

latitude are obtained from computation. Thus, let S be the body's place, 

V the vernal equinox, VD the body's right ascension, I) S its declina- 



r 




PRECESSION AND NUTATION. 37 

fcion, VL its longitude, S L its latitude, and the Fi s- 45 - 

angle L VD the obliquity of the ecliptic. 
Make 

a = VD = right ascension ; 

S = D S = declination ; 

1= VL = longitude; 

X = L S — latitude ; 

w = L VD = obliquity of the ecliptic, 

e = S V = arc of great circle through the 
body and vernal equinox ; 

<p = S VD = inclination of S V to the equinoctial 

Then in the triangle S VD, right-angled at D, 

tan 8 , . 

tan <p = (35) 

sin a v 

cos e = cos a . cos J . . . , . . (36) 

and in the triangle VL 5, right-angled at Z, 

tan / = tan s . cos (<p— w) (37) 

sin X = sin e . sin (9— u) (38) 

§ 153. If the longitude, latitude, and obliquity be given, then in the 

triangle VLS, 

/ v tan X , . 

tan (<p — w) = - — r (39) 

v ' sin I ' 

cos g = cos / . cos X (40) 

and in the triangle V D S, 

tan a = tan s . cos 9 (41) 

sin 8 = sin s . sin <j> (42) 



PRECESSION AND NUTATION. 



§ 154. The longitudes and latitudes of the stars, being thus determined 
at different epochs, show a slow increase in all the longitudes, while the 
latitudes remain sensibly the same. 

§ 155. This is owing to a slow gyratory motion of the line of the ter- 
restrial poles in a retrograde direction, caused by the rotary motion oi 
the earth and the combined action of the sun, moon, and planets upon 
the ring of equatorial matter that projects beyond the sphere of which the 
polar axis is the diameter. It is the resultant of two component motions. 



38 



SPHERICAL ASTRONOMY. 



§ 156. By the first of these components alone, called nutation, the 
line of the poles would describe once in every 19 years an acute conical 
surface, of which the vertex is at the centre of the earth, and the intersec- 
tions with the celestial sphere are two equal ellipses, whose transverse and 
conjugate axes are respectively 18". 5 and 13".74, the former being al- 
ways dire( ted towards the poles of the ecliptic 

§ 157. By the second, called the mean precession, the centres of these 
ellipses are carried uniformly around the poles of the ecliptic from east to 
west in equal circles, of which the radii are about 23° 28', and at a rate 
of 50 ".2 in the interval of time between two consecutive returns of the 
sun to the mean vernal equinox. This interval is called a tropical 
year. The mean equinoxes perform, therefore, one entire revolution im 
360° -f- 50 ".2 = 25817 tropical years. 

In the figure, S is the sun, E the earth, P t the north pole of the ecliptic^ 
P' the true and P the mean north pole of the equinoctial. The curve 
about >S represents the earth's orbit, that 
about E, and of which the plane is perpen- 
dicular to EP', shows the direction of the 
earth's axial motion ; E V is the intersection 
of the plane of this circle with that of the 
ecliptic, and V is the vernal equinox. The 
circle about P t has a radius of about 23° 28', 
the curve about P is the elliptical path de- 
scribed hy the true pole P' about the mean 
P, and of which the longer axis P' P" passes 
through P r The arc V L of the ecliptic, 
the circle about P h and ellipses about P are 
all on the surface of the celestial sphere, while S, E, and the curves aboist 
them, are at its centre. 

§ 158. By the component motions of mean precession and of nutation 
combined, the true equinoctial pole is carried with a variable motion along 
a gently waving curve whose undulations extend to 
equal distances on either side of the circumference 
of mean precession, and which it intersects at points 
separated by angular distances, as seen from the 
centre of the celestial sphere, equal to 19 X 50".2* 
Xsin 23° 28'-^ 2 ± I3".74 = 3'10"± 13",74. 

§ 159. The motions due to the action of the sun 
and moon are opposed to those arising from the ac- 
tion of the planets* and when estimated along the 




Fie: 47. 




PRECESSION AND NUTATION. 39 

ecliptic are called luni-solar precession in longitude. The combined effect 
arising from the simultaneous action of all the bodies, estimated in the 
siame direction, is called the general precession in longitude. 

§ 160. The equinoxes always conforming to the places of the equinoc- 
tial poles, have a slow, irregular, but continuous retrograde motion. 

The place of the vernal equinox without nutation is called the mean 
equinox ; with nutation, the true or apparent equinox. 

The inclination of the equinoctial to the ecliptic without nutation is 
called the mean obliquity ; w T ith nutation, the true or apparent obliquity. 
The difference between the mean and apparent obliquity is called the nu- 
tation of obliquity. 

§ 161. The apparent equinox wanders in either direction from the 
mean to a distance equal to 1 3", 74-^2 • sin 23° 28',^ 17", 25, which it 
reaches when the mean and apparent obliquity are equal ; and the 
apparent obliquity varies on either side of the mean from zero to half 
of 18 // -5 or 9". 25 ; the latter being reached when the apparent equinox 
coincides with the mean. 

§ 162. The motion of the mean equinox along the ecliptic is deter- 
mined by that of the centre of the little ellipse above referred to, and is 
therefore at the rate of 50".2 a year, being the quotient which resulls from 
dividing 360° by the period required for the true pole to perform one 
entire circuit around the pole of the ecliptic. 

§ 163. The distance ffom the mean to the apparent equinox is called 
the equation of the equinoxes in longitude. 

§ 164. The intersection of a declination circle through the mean equi- 
nox with the equinoctial, is called the reduced place of the mean equinox. 

§ 165. The distance from the reduced place of the mean equinox to 
the apparent equinox, is called the equation of the equinoxes in right as- 
cension. 

§ 166. The changes which take place in these equations, as also in the 
apparent obliquity of the ecliptic, are called periodical variations, from the 
circumstance of their running through all their possible values in a com- 
paratively short period. 

Formulas for computing the equations of the equinoxes in longitude 
and right ascension will be given in another place. 

§ 167. Besides the motion of the equinoctial, due to the action of the 
heavenly bodies on the protuberant ring of matter about the terrestrial 
equator, there is another effect due to the deflecting action of the planets. 
By this the earth is turned aside from the path it would describe, if subject- 
ed to the action of the sun alone, and the place of the ecliptic, therefore, 

7 



40 SPHERICAL ASTRONOMY. 

changed. The amount of this change is exceedingly sinal", being onlv 
about 46" in a century. Its present effect is to diminish the mean obli- 
quity, and this will continue to be the case for a long period of ages, when 
the change will be in the opposite direction, the motion being one of oscil- 
lation to the extent of 1° 21' about a mean position. The chauge in the 
value of the mean obliquity arising from the cause here referred to, is 
called the secular variation of the obliquity, because of the great period 
of time required to pass through all its values. 

SIDEREAL TIME. 

§ 168. It has been explained (p. 237) how the motion of the pointers or 
hands of clocks and watches over stationary circular scales of equal parts 
upon their dial-plates, is employed to measure the lapse of time. The uni- 
form motion of the meridian, carrying with it an imaginary movable circu- 
lar scale of equal parts, coincident with the equinoctial, gives the means 
of regulating these and all other artificial time-keepers. 

§ 169. The origin or zero of the equinoctial scale is on the upper me- 
ridian; its unit of measure is one hour, equal to 15°; its pointer or hand 
the declination circle through the centre of some heavenly body, and time 
measured upon it takes the name of the body which regulates the pointer. 

§ 170. The distance of the pointer from the origin or upper meridian, 
estimated westwardly, is the hour angle of the body which gives the scale 
its name, and measures the time since its meridian passage. 

§ 171. Time measured by the hour angle of the mean equinox is 
called mean sidereal time ; and the interval of time between two consecu- 
tive passages of the meridian over the mean equinox, is called a sidereal day. 

§ 172. Time measured by the hour angle of the apparent equinox is 
called apparent sidereal time ; and the interval of time between two con- 
secutive passages of the meridian over the apparent equinox, is called an 
apparent sidereal day. 

§ 173. Apparent sidereal time is that usually employed by astronomers. 
It is affected by the equation of the equinoxes in right ascension, of which 
the value in time being applied to the apparent sidereal time, with its 
proper s>ign, gives the mean sidereal time. This difference between appu 
rent and mean sidereal time is called also the equation of sidereal time. 

§ 174. Apparent sidereal days are slightly unequal; but the fluctua- 
tions of a clock marking apparent from one noting mean sidereal time 
would be only about 2 S .3 in nineteen years. 

§ 175. A timepiece whose hour-hand passes unif>rmly over the circular 



THE EARTH'S ORBIT 41 

scale of 24 hours on the dial-plate, in a sidereal day, is said to run with 
sidereal time ; it will mark mean sidereal time when its hands indicate at 
any and every instant the hour angle of the mean vernal equinox. 

§ 176. The sidereal time of the meridian's passing the centre of any 
body is the true right ascension of the body ; and the rate of the time- 
piece on sidereal time, its error at any epoch, and the indication of the 
hands on its dial-plate at the instant the meridian passes the centre of any 
body, are the data which make known the body's right ascension. 

§ 177. The sidereal day is shorter than the time required for the earth 
to turn once about its axis by about t Jq of a sidereal second. 

THE EARTH'S ORBIT. 

§ 178. The orbit of the earth is an ellipse, of which the sun occupies 
one of the foci. 

§ 179. The extremities of the transverse axis of the orbit are called 
the Apsides ; that most remote from the sun is called the higher and that 
nearest to the sun the lower apsis. The lower apsis is also called the pe- 
rihelion and the higher apsis the aphelion. The transverse axis produced 
both ways is called the line of the apsides. 

§ 180. The place of the sun or other heavenly body which has the 
greatest distance from the earth is called the apogee, and that which has 
the least distance is called the perigee. When, therefore, the earth is in 
aphelion, the sun is in apogee; and when the earth is in perihelion, the 
sun is in perigee. 

§ 181. The quotient obtained from dividing the circumference of a 
circle, of which the radius is unity, by the interval of time between two 
consecutive returns of a body to the same origin, is called the body's mean 
motion from that origin. 

Thus, let T be the interval, and m the mean motion ; then will 



2rr 



(43) 



§ 182. The origin may be movable or fixed ; when in motion, the mo- 
tion may be direct or retrograde. 

§ 183. Denote by r the radius vector of the earth, by c the area which 
this line describes in a unit of time, and by n the true motion, tl en will, 
Analytical Mechanics, equation (266), 

V*5r (44) 



42 



SPHERICAL ASTRONOMY, 



§ 184. The interval of time between two consecutive returns of the 
sun 10 the vernal equinox, is called a tropical year. That between two 
consecutive returns to the mean vernal equinox, a mean tropical year. 

§ 185. The arc of the ecliptic from the mean vernal equinox to the 
place the sun would occupy, had his motion in longitude been uniform 
and equal to a mean of his actual motions, is called his mean longi- 
tude — the true and mean places always coming together on the line of 
the apsides 

§ 186. The interval of time between two consecutive returns of the 
earth to the perihelion or aphelion is called an anomalistic year. 

§ 187. The mean motion of the earth from perihelion is the value of 
ira, in equation (43), the value of I 7 therein being the anomalistic year. 

§ 1S8. The angle ESP, which the radius vector of the earth makes at 
any time with the line of the apsides, reckoned from perihelion, is called 
the true anomaly. 

§ 189. The angle which the radius vector of the earth at any time 
would make with the same line, and estimated from the same point, had 
the earth moved from perihelion with its mean motion, and retained this 
motion unaltered, is called the mean anomaly. 

§ 190. The relaiiou which connects the mean with the true anomaly 
is, Appendix No. V., equation (g), 

n = V— 2 e sin V -f- § e 2 sin 2 V — &c. ... (45) 

in which n is the : lean anomaly, V the true anomaly, and e the eccen- 
tricity. 

§ 191. The diftl fence between the mean and the true anomaly is called 
the equation of the centre. Denoting the equation of the centre by E, we 
have, equation (45), 

E= n — V = - 2esin V+ \e* sin 27- &c. . . (46) 



§ 192. Let S D S. S a S„ represent the 
ecliptic ; S v S a the line of the equinoxes; 
S, S w the line of the solstices ; S v the 
vernal equinox ; S the sun ; P E A E a P 
the earth's orbit • P the perihelion ; A 
the aphelion. 

When the earth is at E v the sun will 
appear at the vernal equinox S v ; wmen 
at E t , the sun will appear at the summer 
solstice S M ; and when at E„ the sun will 




THE EARTH'S ORBIT. 43 

appear at the autumnal equinox S a ; and when at E- w the sun will appear 
at the winter solstice S w . 

§ 193. Let E t be the place of the earth, E m its mean place; then will 
S v E' t , estimated in the order of the signs, that is, in the direction indi- 
cated in the figure, be the earth's longitude as seen from the sun ; S v E' m , 
estimated in the same direction, its mean longitude ; S v P' the longitude 
of the perihelion ; P' E' t the true anomaly ; P ' E' m its mean anomaly, and 
E' t ,S E' m the equation of l the centre. 

§ 194. It is obvious that the equation of the centre is equal to the dif- 
ference between the mean and true longitudes from the same equinox. 

§ 195. The earth's orbit is known when its semi-transverse axis, its ec- 
centricity, and the longitude of its perihelion are known, its plane being 
that of the ecliptic. These are called the elements of figure. The periodic 
time, the mean motion, and the mean longitude at some particular epoch y 
are the additional data from which result by computation the earth's true 
motion and actual place at any other epoch before or after. These are 
called the elements of place and motion. 

§ 196. Make 

L = mean longitude of the earth at the given epoch ; 

t = an interval of time before or after ; 
m = mean motion ; 

a = true longitude at time of observation ; 
a p = longitude of the perihelion : 

then, Appendix No. V., equation (i), % 

L + m t — a — 2 e sin (a — a p ) + | e 3 sin 2 (a — a p ) — &c. (41) 

The sun will have the greatest apparent diameter when the earth is in 
perihelion, and legist when in aphelion ; denote these diameters by S t and 
<T respectively, and the corresponding radii vectors by r t and r' ; then 
from the principles of optics, 

r, : r> : : 5' : * it 
and 

whence 

r '_ r . 8-S' 



r'+r { 6,+ 8' 

Actual measurements ffive about 



s 



5 J = 32',5 
£' = 31'.5 



4A 



SPHERICAL ASTRONOMY. 



whence 



64 



0.016 nearly; 



(49) 



from which it appears that e is so small as to justify the omission from 
equation (47) of those terms in which its powers higher than the first 
enter, and we may write 

L 4- m t = at, — 2 e sin (a — a p ) (48) 

§ 197. From four observed right ascensions of the sun, compute, by 
equation (33), his corresponding true longitudes; each longitude increased 
by 180° will give the corresponding true longitude of the earth ; denote 
these by a u a^ a 3 , and a 4 , and the intervals of time from the epoch of the 
mean longitude L, say noon, January 1st, to the times of observation, by 
t u 4* *3, aQ d t 4 respectively, then will, equation (48), 

L + m t x = a L — 2 e sin (a, — a^,), 
L + m t 2 = a 2 — 2 e sin (a 2 — a^,), 
L + m t z = a 3 — 2 e sin (a 3 — a p ), 
L + m t A = a 4 — 2 e sin (a 4 — a^), ^ 

four equations, from which the mean longitude L at the epoch, the mean 
motion m, the eccentricity e, and longitude of the perihelion a p , may be 
JfounoV For this purpose subtract the first from the second, 

m(t 2 — ti) = a 2 — aj — 2 e [sin (a 2 — a p ) — sin (a, — a p )] ; 

making 

t 2 — t x = d, a 2 — a x = a, or a 2 = a t + #> 

and reducing by the relation 

sin (a 2 — a p ) = sin (a,— a p -{- a) = sin (a,— a ? ) cos a + cos (a a — a p ) sin a, 

we find 

m& = a -{- 2 e [sin (a, — a p ) (1 — cos a) — cos (a! — a p ) sin a] (50) 

Hubtracting the first of equations (49) from the third and fourth, making 

h — ^i = o , £j — t x = o ; 
cc 3 — a, = a', a 4 — a t = a" ; 

reducing in the same way, and replacing 1 — cos a by its equal, 2 sin 8 ^ a, 
we find, including the equation above, 

2 e [2 sin (a t — a p ) sin 2 i a — cos (a t — a p ) sin a J, 



md —a 

mb' — a' = 2 e [2 sin (a! — a ) sin 2 \a' — cos (a x — a^,) sin a ], 

md"— a" = 2 e [2 sin (a, — a p ) sin 2 \ a" — cos (a t — a ? ) sin a"]. 



THE EARTH'S ORBIT. 45 

Dividing the first of these by the second and third successively, making 

mQ — a __ ") 

= M, 



mb' — a' ' I 

V (51) 

= JST: 



m r-a"~~ ' J 



and dividing both numerator and denominator of the secorid members by 
cos (a, — - u p ), we have 



if = 



2 tan (a, — a p ) . sin 2 \a — sin a 
2 tan (a, — a p ) . sin 2 A «' — 



>; .am -2 

_ 2 tan (a t — a^,) sin 2 ^ a — sin a 
2 tan (a, — a p ) sin 2 -i- a" — sin a' 



(52) 



from which we find 

. M sin a'— sin a JV* sin a" — sin a 

2 tan (a, — a p ) ±= — . . . = •= _ . . - ..- r^-y- (53) 

v p/ ifsin 2 ^a'— sm 2 la i^sin 2 \ a"— sin 2 A a v 7 

in which the only unknown quantity is m; this entering, equations (51), 
the values of M and N. 

To find the value of m, clear the fractions, transpose to the first mem- 
ber, and make 

n = sin a sin 2 ^ a' — sin a' sin 2 \ a 1 
Jc 3= sin a' sin 2 J a"-— sin a" sin 2 ^ a', 
t 2= sin a" sin 2 J a — sin a sin 2 -J a", 

or reducing for the sake of logarithmic computation by the relation 

sin a = 2 sin J a . cos A. a, 

n = 2 sin \ a sin A a' . sin \ (a! — a ), ) 

& = 2 sin i a' . sin £ a" . sin \ {a"— a'), V . . (54) 

e = — 2 sin 1 <x sin A a" . sin £ (a"— a) : ) 

we have 

Jf » + iV^ 4. & . M. N = 0. 

Replacing if and iv*by their values given in equations (51), we find 

n (m — a) i (m & — a) k (in & — a) 2 

mt'-a' + md"-a" + (m d'- a') (w &"- a") = ' 

whence 

(m 6 — a) [ (n &" + i&' -{- k d) m— (n a" + i a' -f- * a) ] = 0. 

But m 6 — a cannot be zero, since e is not zero. 



46 SPHERICAL ASTRONOMY. 

Placing, therefore, the second factor equal to zero, we find 

« H — a H — « 

no' + »o' + «a n n 

W = n4"+i«' + Al = " r F~ ' ' ' (55) 
6» + _ d' + -6 
n n 

From equations (54) we have 

i sin \ a" . sin ^ [a" — a) | 



« sin \ a' . sin \ («' — a)' 

& sin ^a" . sin ^ (a" — a') 

| a . sin 1 (a' — a) 1 



(56) 



» — 2 



Now a, a', and a" are the increments of the true longitude since the 

i k 

first observation; these in equations (56) give the fractions — and — ; 

these in equation (55) give the value of m; this in equations (51) and 
(52) give the value of tan (oti — a p ) and therefore of a p ; this, in equation 
(50), gives the value of e, and this, together with m and a p , in first of 
equations (49), gives the value of L. 

'§ 198. The mean motion in longitude, the eccentricity and longitude 
of the perihelion being determined at dates remote from one another, are 
found to be very slightly variable. The present value of the eccentricity 
is 0.01678356, the semi-transverse axis of the earth's orbit, or the earth's 
mean distance from the sun being unity ; that of the mean motion in lon- 
gitude in one sidereal day is 0°. 98295603 ; the longitude of the perigee 
at the beginning of the present century was 279° 30' 05".0, and the 
mean longitude of the sun at the same time was 280° 39' 10". 2. 

The longitude of the perihelion is found to increase at a mean rate of 
'61 ".9, in a tropical year, and deducting 50".2 for the retrocession of the 
mean equinox, gives to the perihelion a direct motion of 11". 7 through 
space in the same time. 

§ 199. Denoting by y u the length of the tropical year in sidereal days, 

we have 

360° 360° , , M 

y t = = — = 366.242 days . . . (57) 

y '* m 0°.98295603 J y J 



MEAN SOLAR TIME. 

§ 200. Although the mode of reckoning time by the motion of the ver- 
nal equinox affords great facilities in practical astronomy, it is of little or 
no use in the ordinary operatic ns of common life. Business and social in- 



MEAN SOLAR TIME. 4.7 

tercourse are mostly regulated by the alternations of daylight and darkness, 
and the sun is the natural object of reference in all divisions of time for so- 
ciety in general. 

I 201. Time measured by the hour angle of the sun is apparent solar 
time. 

§ 202. The epoch of the sun's being on the meridian of a place, is 
called apparent noon of that place. 

§ 203. The interval of time between two consecutive passages of the 
sun's centre over the upper or lower meridian of the same place, is called 
an apparent solar day. 

The apparent solar is longer than the sidereal day, in consequence of 
the earth's real, and therefore of the sun's apparent, motion in the ecliptic 
in an easterly direction. If, for instance, the vernal equinox and the sun 
were to pass the meridian of a place at the same instant to-day, the sun 
would be to the east of the equinox on the morrow, and would cross the 
same meridian after it. 

§ 204. The orbital motion of the earth and, therefore, the apparent mo- 
tion of the sun in the ecliptic is, Eq. (<*), Appendix V, variable. The 
unequal arcs which measure the daily increments of the sun's longitude 
vary their inclination to the equinoctial from about 23° 28' at the equi- 
noxes, to zero at the solstices ; and these unequal arcs may hence be pro- 
jected by declination circles into still more unequal arcs of right ascension. 
These latter measure the excess of the different apparent solar over the si- 
dereal days ; and hence the variable orbital motion of the earth, and the in- 
clination of the plane of its orbit to the equinoctial, conspire to make the 
lengths of the apparent solar days unequal. 

§ 205. Timepieces cannot be made to imitate this inequality, nor is it 
desirable they should do so, were it possible. 

Had the earth's orbit been circular and in the plane of the equinoctial, 
its orbital motion would have been uniform, the sun's apparent daily in- 
crease of right ascension constant, and the apparent solar days of equal 
duration. 

§ 206. These conditions are fulfilled by the device of an imaginary sun 
conceived to move uniformly in the equinoctial with the true sun's mean 
motion in longitude, and to set out from the reduced place of the mean 
vernal equinox when the true sun's mean place leaves the mean equinox. 

This imaginary body is called the mean sun. 

§ 207. Time measured by the hour angle of the mean sun is called 
mean solar time. The epoch of the mean sun being on the meridian of a 
place, is called mean noon of that place. 



■ 



48 



SPHERICAL ASTRONOMY. 



Fig. 49. 




§ 208. The difference between the apparent and mean solar time is 
called the equation of time. If to the mean time the equation of time be 
applied with its proper sign, the apparent time will result ; if the equation 
of time be applied with its proper sign 
to the apparent time, the mean time 
will result. 

The equation of time is employed to 
pass from mean to apparent, or from 
apparent to mean time. 

§ 209. Thus, let P M be an arc of 
the meridian, VM of the equinoctial, 
V E of the ecliptic ; P the pole of the 
equinoctial ; V the true, V m the mean, 
and V r the reduced place of the mean 

equinox ; S the true and S m the mean sun ; then will MP S be apparent, 
and MP S m mean solar time ; VS a the right ascension of the real sun ; 
V V r the equation of the equinoxes in right ascension. 

Make 

e = S a S m z= the equation of time ; 
a =z VS a = the right ascension of true sun ; 
I = V T S m = the mean longitude of the sun ; 
q = VV r = the equation of the equinoxes in right ascension ; 

then from the figure, we have 

e = a - (I + q) (58) 

that is, the equation of time is equal to the sun's true right ascension di- 
minished by the sun's mean longitude, corrected for the equation of the 
equinoxes in right ascension. 

§ 210. When the sun's true right ascension exceeds the corrected mean 
longitude, the equation of time must be added to apparent time to obtain 
mean time, and vice versa. The equation of time is zero four times a year, 
vjg., on 15th April, 14th June, 31st August, and 24th December. 

§ 211. The mean sun and mean equinox when together must pass some 
meridian at the same instant. When the same meridian returns to the 
mean equinox on the following day, the mean sun will be to the east by a 
distance equal to that which measures its motion in one sidereal day ; and 
the mean solar day will exceed the sidereal day by the interval of sidereal 
time required for the meridian to overtake the mean sun after it passes the 
mean equinox. 

Denote this excess by t, expressed in days ; and the motion of the mean 



M 

MEAN SOLAR TIME. 49 

sun in one sidereal day, equal to the earth's mean orbital motion in the 
same time, by m. Then will m t be the motion of the mean sun in the 
time £, and its right ascension from the mean equinox at the instant the 
meridian overtakes it will be m -f- »* & But this is the hour angle of the 
mean equinox, or the sidereal time t, reduced to degrees ; whence 

m + mt = 3QQ°X t\ 
or «^L- 

M 



360 — m 1 
and for the length oTthe mean solar day, expressed in sidereal time, 

or replacing m by its value 0°.98295603, § 198, and denoting the length 

of the mean solar day by D m expressed in terms-of the sidereal day D t , as 

unity, we have 

Z> w = 1.00273791 2), (59) 

and 

A = £-z — = 0.99726957 D m , 

s 1.00273791 

Whence to convert intervals of mean solar into intervals of sidereal, or in- 
tervals of sidereal into intervals of mean solar time, we have these rules, 

viz.: 

Sidereal interval = 1.00273791 X Soldr interval, 

Solar interval = 0.99726957 X Sidereal interval. 

§ 212. Applying this second rule to the length of the tropical year ex- 
pressed in sidereal days, we have, Eq. (59), 

Solar interval = 0.99726957 X 366.242 == 365.2422414; 

or reducing the fraction to hours, minutes, and seconds, and denoting the 
length of the tropical year, expressed in mean solar time, by y m , we have 

y m = 365 d 5 fe 48 m 4^ ...... (60) 

§ 213. Denote by y^ the length of the anomalistic year expressed in 
mean solar time; then, §§ 157 and 198, 

360° - 50 ; ' '.2 : 360° -f 11".7 : : 365^ 5 fe 48 w 48 s : y^; 
whence 

#«« = 365* 6 h 13 CT .3 (61) 

§ 214. The interval of time required for the earth to perfcrm one entire 
circuit about the sun in space is called a sidereal year, 

4 



50 SPHERICAL ASTRONOMY. 

Denote by y sm the length of the sidereal year in mean solar time, then 

360° - 50".2 : 360° : : 365 d 5 n 48 m 48 s : y im ; 

whence 

y m = 365 d 6 h 9 m 9 8 .6 (62) 



ABERRATION. 

§ 215. The earth's orbital motion, combined with the motion of light, 
produces an apparent displacement of all the heavenly bodies in the di- 
rection of the point of the celestial sphere towards which the earth is, at 
the instant, moving. This displacement is called aberration. 

Thus, let S be the place of a heavenly Fig. 50. 

body, i^that of the earth moving from M s„S1L—§' 

towards JV" along an arc of its orbit. / j / 

From E take any distance E ' E' ' ; join S 1/ ! 

and E', and lay off upon E' S the distance J_ /*, 

W C y which bears to EE\ the ratio of A J 

the velocity of light to that of the specta- // / 

tor, and suppose connected like himself / / / 

with the earth. / / / 

Now a wave of light from S, another / 1/ 

which originates at C, when that from S E E' 

passes this point, and the spectator's eye 

starting from E at the same instant, will all meet at E\ and as bodies al- 
ways appear in the direction of the normal to the wave front, the point C 
and the body S will be seen in the direction E'S. But C, having a ve- 
locity equal and parallel to the spectator, will have passed on to C, at the 
extremity of a line through C equal and parallel to E ' E' ' ; so that when 
C appears in the direction of the body S, it will, in fact, be in advance of 
it by the angle SEC. 

Let C be the optical centre of the object-glass of a telescope, attached 
to the face of a graduated circle, moving in the plane of the body and the 
tangent line to the terrestrial orbit at the earth's place, and E' the inter- 
section of the cross wires at the solar focus ; then, when the image of the 
body appears at the latter point, the line of collimation will be in advance 
of the body itself, and its instrumental bearing will be in error by the 
angle S E' C, and must be corrected by the same angle to get the true 
bearing. 

g 216. But had S been a terrestrial object, by the t'nie its light fix in 



ABERRATION". - 51 

the positi-on S had reached C, the body itself would have been at S", the 
intersection of E C produced and S S' drawn parallel to E E' ; and at 
the instant of its li<rht reaching E' the body would have been at S\ the 
intersection of £ S" produced and the line of collimation. Geodetical 
observations are, therefore, unaffected by aberration, while astronomical 
observations are, in general, affected by it. 
| 217. Make 

r=$E'S' = ECE f = aberration ; 

a = S E' N = angle the direction of the body makes with 

that of the earth's motion. 
V = velocity of the earth ; 
V = velocity of light : 

Then, in the triangle C E' F^ 

V : V : : sin (a — r) : sin r, 

whence 

V 

sin r = — . sin (a — r) . . . . * . (64) 



If p, denote the mean radius of the earth's orbit, then will 

V: 



2 tfp, 



305 d .25636 ' 

and it will be shown hereafter that light requires 16™ 26* to pass over fhe 
distance 2 p y , and therefore 

1 F ~ lG' n 26" " 

whence 

F 3.1 41 G X 16 m 26' 



F 365 d .25636 



= 0.00009815, 



from which, and equation (64), it is apparent that r is very small, and 
may be neglected in comparison with a ; we may therefore write 



200264/'.8 



= 0.00009815 sin a, 



) ^ *'6 <s>L**4 



which 206264.8 is the number of seconds in radius ; whence 
r"= 0.00009815 X 206264".8 sin a, 

r"=20".246sina. ......... (65) 



< 



52 SPHERICAL ASTRONOMY. 

§ 218. Let AB be the intersection of the -celestial Fig. 51. 

sphere by a plane through the body and the direction r<^~ ^~~Sf 
of the earth's motion, A C that of a plane through I X^-''' / 
the observer and star, and perpendicular to the plane j ^/ \ / 

of the ecliptic, and B C an arc of the ecliptic ; then g -^ 

will B be the point in which the tangent to the earth's 
orbit at the place of the earth pierces the celestial sphere, A will be the 
projection of the body upon the celestial sphere, and AB = a; and if 
A C = X and C A B = 9, we have 

cos 9 = tan X cot a, 



and 

whence 


cos 2 9 = 
1 — sin 2 9 = tan 


tan'X 


. cot 2 a, 

— sin 2 a 

sira 






and solving with 


respect to sin a 7 












tan X 




tan X 






and therefore, 


Vl + tan 2 X - 


sin 2 9 


Vsec 2 X — 


sin 8 


— 3 




sin a — — ■ — 


sin X 








Vi 


— cos 2 


7 — ~2~~ 7 

X . s\w 9 





and this in equation (65) gives - 

20".246 sin X 



yl — cos 2 X . sin 2 9 



(66) 



which is the polar equation of an ellipse, the pole being at the centre. 
So that, if the image of a fixed star were kept constantly on the cross wires 
of a telescope during one entire revolution of the earth in its orbit, the 
line of collimation would trace upon the celestial sphere ^an ellipse of which 
the star would occupy the centre; the semi-transverse axis would be 
20 ".24 6 and the eccentricity cos X. 

If the star were in the plane of the ecliptic, then would X = r 
cos X = 1, and the orbit would become a right line equal in. length to 
40".492. If the star were at either pole of the ecliptic, then would 
X = 90, cos X = 0, and the orbit would be a circle. Between these limits 
the eccentricity will vary from 1 to 0. 

The coefficient 20".246 is called the constant of aberration, ^ 
§ 219. Since the aberration is in the arc AB, its projection on AC 
will be the aberration in latitude. Denoting the latter by r\ we have 






HELIOCENTRIC PARALLAX. 53 

20".246 . sin X . cos <p , , 

r' = —. (67) 

V 1 — cos 2 X. sin 2 9 

which is obviously the greatest when 9 = 0° or 180°, in which case the 
earth will be moving parallel to the circle of latitude of the body, and the 
aberration in latitude will be equal to 20".246 • sin X, which is the 
semi-conjugate axis of the ellipse. 
The aberration in longitude denoted by r, will give 

20 ff .246 .tanX.sin 9 , . 

r ' = /, *-r =^ (68) 

V 1 — cos X . sin 9 

which is the greatest when <p = 90° or 2*70°, in which case the earth will 
be in the act of passing the body's circle of latitude, and the corresponding 
aberration will be 20'\246, the semi-transverse axis of the ellipse. 

§ 220. Equations (66), (67), and (68) are applicable to a body which 
has no proper motion of its own. In case the body has a motion, this 
must be allowed for in clearing its instrumental bearing of aberration, and 
the mode of doing this will be indicated under the head of planets. 

§ 221. In the case of the sun, which may be regarded as fixed, 9 is 
always 90°, sin 9 = 1, and replacing 1 — cos 2 X by sin 2 X, equation (66), 
reduces to 

r = 20".246 ; 

that is, the sun will always appear behind his true place by the constant 
of aberration. 

§ 222. In conclusion, it is proper to remark that V, the velocity of the 
earth in its orbit, which is assumed to be constant, is not strictly so, but 
the variation is so small as not sensibly to affect the foregoing results. 
The actual velocity varies inversely as the length of the perpendicular 
drawn from the sun to the line which is t'-ingent to the earth's orbit at the 
earth's place {Analyi. Mechanics, g 19-3). But the eccentricity of the 01 bit 
being very small, gives but little variation in this perpendicular. 



HELIOCENTRIC PARALLAX. 

§ 223. The place in which a body would appear if viewed from the 
centre of the sun, is called its Heliocentric place. 

§ 224. The arc of a great circle of the celestial sphere drawn from the 
heliocentric to the geocentric place of a body, is called its Heliocentric 
parallax ; and is obviously the path a body would appeal' to describe to 



54: 



SPHERICAL ASTRONOMY. 



an observer were he to pass from one extiemity to 
the other of the earth's radius vector. 

Thus, let S be the sun, E the earth, B the body, 
and MN the inte; section of the celestial sphere 
by a plane through the body and radius vector 
SE; then will B' be the heliocentric, and B n 
the geocentric place of the body; and B" B' its 
heliocentric parallax. The helioeentiic parallax 
measures the angle B" BB' = S B E = the angle 
at the body subtended by the radius vector of the 
earth. 

§ 225. Make 

D = S B = distance of body from sun ; 

R= SE == earth's radius vector; 

r / = S BE= heliocentric parallax; 

a / = SEB = angular distance between sun and body ; 

then in the triangle SEB, 

D : R : : sin a. i r sin r lt 
whence 

R 




sin r i = —■ . sin a t 



(my 



When a t = 90°, then will r t be the greatest possible. This maximum 

heliocentric parallax, is called the cmnuat parallax ; which denote by **, 

and we have 

R 



sm * = — 



and if nt be very small,, 



« is expressed in seconds, and 
From this we obtain 



«_R 

denotes the number of seeonds in radius. 



D-R 



m 



This gives the distance of the body from the sun in terms of its annua' 
parallax and the earth's radius vector. 

§ 226. Substituting the value of D in Eq. (69), and making 



me have 



r. = if . sin a, 



(71) 



HELIOCENTRIC PARALLAX. 



55 



§ 227. Let S be the sun's place in the eclip- 
tic, B the place of the body, BA the arc of a 
circle of latitude, S A an arc of the ecliptic, 
and E the earth. The side SB will measure 
the angle a, ; and denoting- the side A B by X, 
and the angle SB A by <p ; , we have 

cos <p y — tan X . cot a y ; 

and bv a transformation, the same as in § 218, 



Fig. 53. 




sin X 



which in Eq. (71) gives 



V 1 — cos 2 X . sin 2 <p 



<jr . sin X 



V 1 — cos 2 X . sin 2 <p ; 



(72) 



This is the polar equation of an ellipse having the pole at the centre ; 
and it shows that the parallactic path of a body's geocentric place, due to 
the earth's orbital motion alone, is an ellipse of which the centre is the 
body's heliocentric place. 

The semi-transverse axis and eccentricity are respectively if and cos X. 
If the body be in the pole of the ecliptic, then will X = 90, cos X = 0, 
and the ellipse becomes a circle ; if in the ecliptic, then will X = 0, cos X 
= 1, and the ellipse becomes a right line whose length is 2 k. 

§ 228. Heliocentric parallax throws a body from its heliocentric place 
towards the geocentric place of the sun or towards that point in which the 
earth's radius vector, produced beyond the sun, pierces the celestial sphere. 
Aberration throws it towards the point in which the tangent line to the 
earth's orbit, at the place of the earth, pierces the same surface. Both 
points are in the ecliptic, and if we neglect the eccentricity of the earth's 
orbit, which we may do without sensible error when the heliocentric par- 
allaxes are. employed, these points are 90° apart. When, therefore, <p = 0° 
in Eq. (66), then will <p y = 90° in Eq. (72), and vice versa ; and the least 
possible heliocentric parallax will occur at the time of the greatest aberra- 
tion, and the least aberration at the time of the annual parallax. 

§ 229. When the longitudes of the sun and body diner by 90° or 270°, 
the heliocentric parallax will become the annual; and if the longitudes and 
latitude of the body be taken at these times and cleared from the effects 
of aberration and nutation, there will result the longitudes and latitudes of 
two points separated by 2 if or double the annual parallax. The value of 



56 SPHERICAL ASTRONOMY. 

*r then becomes known by a sir. sple proposition in spherical geometry, and 
substituted in Eq. (70) gives the body's distance from the sun. 

§ 230. The geocentric co-ordinates of a body corrected for heliocentric 
parallax, become the heliocentric co-ordinates, that is, the co-ordinates as 
they would appear if viewed from the centre of the sun. 

THE SEASONS. V 

§ 231. The sun is the great fountain of those ethereal undulations which, 
acting upon the material of the earth's crust, give to the latter its surface 
heat ; and the temperature of a place depends upon its exposure to their 
calorific action. While the sun is above the horizon, the place is receiving 
heat, and while below, parting with it ; and in such proportion that the 
whole quantity gained and lost balance each other, since every location 
has nearly a constant average of annual mean temperature, as indicated by 
the thermometer. 

§ 232. Whenever the sun is above the horizon more and beneath less 
than twelve hours, the general temperature of the place will be above the 
average, and the converse. 

§ 233. A portion of the wave having a front surface equal to unity can 
generate but a limited quantity of heat, and, all other things being equal, 
the temperature at any one location will be inversely proportional to the 
extent of the' earth's surface upon which this unit is made to act. If the 
wave front be parallel to the earth's surface the temperature will be great- 
est, for then the action is confined to the narrowest limits ; if very oblique, 
the temperature will be low because the action is diffused over a larger 
space. 

§ 234. Let AB be the section, by a vertical plane through the sun's cen- 
tre, of a portion of the wave front, the surface of this portion being unity, say 
ten square miles ; and A C the projec- 
tion of the same on the earth's sur- 
face by normals to the wave front, jk 
called rays. B f L 

The sections are sensibly rectilinear / ^\. k 

within the limits assumed, and the / ^^v 

rays being normal to the wave front, - ^^^^^^^^^^^ 
make with the line of the zenith and 

nadir to the earth's surface, an angle equal to B A C, equal to the sun's 
zenith distance, which being denoted by z, we have 

A C = A B . sec z. 



THE SEASONS. 



« 



Denote by /' the temperature when the wave and eartl^ surfaces are par- 



allel, and by / when they are oblique ; then 

AJB .secz : AB :: T : I 



whence 



sec z 



cos z\ 







and if I t denote the temperature which would result at the unit's distance 
from the sun, and r the radius vector of the earth, we have from the law 
of diffusion, depending upon distance, 



/' = 



I 



whence 



'-% 



cos z 



(13) 



d % in which d de- 



§ 235. Resuming Eq. ( 6 ), and making p = 90 c 
notes the sun's declination, we have 

cos z — sin I . sin d + cos I . cos d . cos P . . . (74) 

which, in Eq. (73), gives 

I = -j . [sin I . sin d + cos I . cos d . cos P] . . (75) 



This result is wholly independent of terrestrial longitude, and is only de- 
pendent on the latitude of the place, the sun's declination, and the place 
of the earth in its orbit. All places upon the same parallel are equally 
exposed, therefore, to the solar influence, and whatever differences of mean 
temperature and of climate they may exhibit are due to local causes, such 
as the vicinity of mountains, extended plains, forests, deserts, or large 
bodies of water, upon all of which the sun is known to produce great va- 
riety of thermal effects. 

§ 236. Makings = 90°, in Eq. (74), we have 



cos P = — tan / . tan d 
and making P = 0, in Eq. (75), we have 

I = -~ cos (I — d) 



(«) 



("> 



Eq. (76) gives the value of the semi-upper diurnal arc, or the time the sun 
is above the horizon, or the duration of calorific action; and Eq. (77) the 
intensity of the solar influence when greatest. 



58 



SPHERICAL ASTRONOMY. 



§ 237. In the course of the tropical year the declination varies nearly 
47°, the sun being at one time about 23°. 5 north, and at another about 
the same distance south of the equator. 

As long as the latitude and declination are of the same name, that is, 
both north or both south, the sun will, Eq. (76), be longer than twelve 
hours above the horizon, and the place will receive more heat than it loses. 
And in proportion as the latitude and declination approach to equality, 
the intensity of the solar action will, Eq. (77), approach its maximum. 
This periodical variation in the daily average temperature of a place, 
caused by a change of the sun's declination, gives rise to the phenomena 
of the seasons. 

§ 238. The interval of time during which the daily increment of tem- 
perature of a place is increasing is called its spiking ; that during which 
this increment is decreasing is called its summer ; that during which the 
daily decrement is increasing is called its autumn or fall ; and that during 
which this decrement is decreasing is called its winter. 

§ 239. Within the tropics C C and 
DD', and especially about the equator 
Q Q\ the temperature is, Eqs. (76) and 
(77), nearly uniform, and always high. 
On this account the terrestrial belt 
bounded by the tropics is called the 
torrid zone. 

Between the tropics and polar cir- 
cles A A' and B B' the average daily 
temperature is much less uniform and 
always lower than in the torrid zone. 
The belts bounded by the tropics and 
polar circles are called temperate zones. 

. Between the poles P and P' and polar circles, the variation of the av- 
erage daily temperature is the greatest possible and the temperature itselt 
least. The portions of the earth's surface about the poles and bouuded by 
the polar circles are called frigid zones. 

§ 240. Places within the torrid zone may be said to have two of each 
of the seasons during a tropical year, and all places in the temperate and 
frigid zones but one. 

For all places in the north temperate and frigid zones, spring begins 
when the sun is on the equator and passing from south to north, or on the 
20th March ; summer, when the sun reaches the tropic of Cancer, or on 
the 21st June; autumn, when the sun returns to the equator in passing to 




TRADE WINDS. 59 

the south, or 22d September ; and winter, when the sun reaches the tropic 
of Capricorn, or 21st December. For all places in the south temperate 
and frigid zones the names of the seasons will be reversed — spring becomes 
autumn, and summer winter. 

§ 241. The elliptic form of the earth's orbit causes the radius vector, 
and therefore, Eq. (77), the intensity of the solar heat, to vary. But the 
angular velocity of the earth about the sun also varies, and according to 
the same law, viz. : that of the inverse square of the earth's distance from 
the sun — Analytical Mechanics, Eq. (266). Equal amounts of heat will 
therefore be developed while the earth is describing equal arcs of longitude, 
and the supply will be the same during the description of any two seg- 
ments, equal or unequal, into which the entire orbit is divided by a line 
through the sun. The earth is nearer the sun while the latter is south of 
the equinoctial, or from the latter part of September to the latter part of 
March ; and it describes the corresponding part of its orbit in a time so 
much shortened as just to balance the increase of thermal intensity. But 
for this law of compensation, the effect would be to increase the difference 
of summer and winter temperature in the southern and to diminish it in 
the northern hemisphere. As it is, however, no such inequality is found 
to subsist, but an equal and impartial distribution of heat and light is ac- 
corded to both hemispheres. 

§ 242. But it must not be inferred that the mean surface heat is con- 
stant throughout the year ; for such is not the fact. By taking, at all sea- 
sons, the mean of the temperatures of places diametrically opposite to one 
another, Professor Dove finds the mean temperature of the whole earth's 
surface in June considerably greater than that in December. This is due 
to the greater amount of land in that hemisphere which has its summer 
solstice in June ; the thermal effect of the sun on land being greater than 
that on water. 

§ 243. The variation of the radius vector amounts to about -^ of its 
mean value, and therefore the fluctuation of heat intensity to about y 1 ^ of 
its average measure — a circumstance which is manifested in a great excess 
of local heat in the interior of Australia during a southern, over that of the 
deserts of Africa during a northern summer. 

TRADE WINDS. 

§ 244. A discussion of the trade winds, the earth's magnetism, and the 
tides, belongs, in strictness, rather to terrestrial physics than to astronomy ; 
but the necessary connection of these phenomena with the earth's diurnal 



60 



SPHERICAL ASTRONOMY. 



rotation and the action of foreign bodies upon the earth, as well as their 
importance to navigation, make a sufficient apology for introducing them 
here. 

§ 245. The surface of the torrid zone is most heated; its excess of 
temperature is communicated to the superincumbent atmosphere; the 
latter is expanded, and becoming specifically lighter, is pressed upward by 
the colder portions on the north and south which move in and take its 
place. These, in their turn, are heated, expanded, and pressed upward, 
and a constantly ascending current is thus produced over an entire zone, 
of which the boundaries fluctuate with the varying declination of the sun 
and the proportion of land and water on the belt of the earth's crust 
lying immediately under the sun's diurnal path. The air thus accumu- 
lated at the summit of the ascending column, being unsupported on the 
north and south, flows off under the action of its own weight in either di- 
rection towards the poles, and, after cooling, descends again to the earth's 
surface in the higher latitudes of the temperate zones to supply the place 
and follow the course of that which has passed to the torrid zone. 

§ 246. Two atmospheric rings, as it were, distinguished by peculiarities 
of internal circulation, are thus made to belt the earth on either side of 
the equator in directions paral- 
lel or nearly so to that great 
circle. On the lower side of 
these rings, in contact with the 
earth, the air moves towards 
the base of the ascending col- 
umn, and on the upper towards 
the poles. 

§ 247. By the diurnal mo- 
tion of the earth, places on the 
equator have the greatest velo- 
city of rotation, and all other 
places less in the proportion of 

the radii of their respective parallels of latitude. The portions of the 
ascending column which flow towards the poles set out with the east- 
ward intertropical velocity, which they carry with them in part to the 
higher latitudes, where they descend to the earth's surface. To an ob- 
server situated in these latitudes, the air will have an apparent east- 
wardly motion, approaching to the excess of the intertropical velocity over 
that of the observer's parallel. Here westerly winds prevail. 

§ 248. On parallels a few degrees lower, the tendency of the air is 




"TfcADE WINDS 61 

towards the equator, and this combined with what remains of the 
apparent easterly component, just referred to, gives rise in the north- 
ern hemisphere to a northwesterly and in the southern to a southwesterly 
wind. 

§ 249. In its onward course towards the equator, this same air crosses 
successively parallels of greater and greater velocity, and this, together 
with friction against the earth's surface, reduces the air's excess of easterly 
motion to zero, and here northerly winds prevail in the northern and 
southerly winds in the southern hemisphere. 

§ 250. In latitu les still lower, the excess of rotation is in favor of the 
earth's surface, and the air, unable to keep up, now lags behind, and ap- 
parently tends to the west ; and here, if the places be in the northern 
hemisphere, northeasterly, and if in the southern hemisphere southeasterly 
winds prevail. 

§ 251. Nearer to the equator the radii of the parallels vary less rap- 
idly, and the velocities of places on the same meridian are more nearly 
equal. In crossing these parallels the air in its onward course finds less 
variation in the velocity of the earth's surface, and friction, which now 
urges the air to the east, together with the easterly pressure below, arising 
from the westerly lagging in the summit of the ascending column, due 
to its decreasing angular motion as it recedes from the centre of rotation, 
soon brings the air and earth to relative rest. This occurs within the base 
of the ascending column where the currents of air, which are continually 
approaching each other from the directions of the poles, meet. This is, 
therefore, a region of calms. 

§ 252. The aerial currents thus produced under the combined influence 
of solar heat and the diurnal motion of the earth, are called Trade winds ; 
and they are so called from the benefits they are continually conferring on 
trade dependent upon navigation. 

§ 253. A voyage from the United States to northern Europe in a 
sailing vessel is on an average ten days shorter than in the contrary direc- 
tion. A sailing vessel on a passage from northern Europe to the southern 
coast of the United States would proceed to the Madeiras to take the east- 
erly trades, and returning would proceed to the Bermudas to catch west- 
erly trades. 

§ 254. Within the region of calms the ascending column of air car- 
ries with it a large amount of aqueous vapor. In its ascent the air expands, 
its temperature is depressed, its aqueous vapor is first condensed into clouds, 
then into rain, and thus the region of calms is also a region of dense 
clouds and copbus rains ; the former giving to the earth, as viewed from 



62 SPHERICAL ASTRONOMY. 

^t 
a distance, the appearance of being girted by dark broken belts, arranged 

in zones parallel to the equator, 

§ 255. The limits of the trades do not always occur in the same lati- 
tudes, but vary with the season. In December and January, when the 
sun is furthest south, the northern boundary of the northeast trades of the 
Atlantic is about 20° N., whilst in the opposite season, from June to Sep- 
tember, it is 32° N. 

§ 256. Owing to the great disparity in the effects of solar heat upon 
land and water, and to the influence of mountain ranges and valleys upon 
atmospheric currents, the regular trades only occur, as a general rule, at 
sea, though in some level countries, within or near the tropics, constant 
easterly winds prevail. This is remarkably the case over the vast plains 
drained by the Amazon and lower Orinoco. 

§ 257. The trades of the ocean and of the land are separated by a 
belt, within which other and variable winds occur. This belt lies upon 
the ocean, and extends along the coasts. When to the east of the trades, 
it is often a hundred miles wide, but when to the west its width is much 
smaller. The interruption of the trades, here referred to, is due to the 
difference of temperature of the air on sea and land, which changes with 
the seasons. The air over the land in the higher latitudes is the warmer 
when the meridian zenith distance of the sun is least, and colder when 
greatest. During the first period the wind is from the sea to the land, 
and in the second from the land to the sea, thus giving rise to the period- 
ical winds called Monsoon*, which occur even within the limits of the 
trades. A large island thus circumstanced is surrounded by a wind blow- 
ing from all quarters at the same time. 

§ 258. A similar difference of temperature, but which varies with the 
alternations of day and night, gives rise to what are called the sea and 
land breezes. 

TERRESTRIAL MAGNETISM. 

§ 259. Another most important effect from the solar heat, combined 
with the diurnal motion of the earth, is the earth's magnetism. 

§ 260. A difference of temperature in different parts of any body form- 
ing a continuous circuit is ever accompanied by electrical waves, propa- 
gated from the hotter to the colder parts. If the circuit be composed of 
various materials, possessing different powers of conducting heat, this differ- 
ence may be maintained in greater degree and duration, and the effects of 
the electrical flow rendered more strikingly manifest. 



TERRESTRIAL MAGNETISM. 63 

§ 261. When the source of heat is moved gradually along the circuit, 
the electrical flow is in the direction of this motion, the colder portions 
always lying in advance and the warmer behind the moving source. 

§ 262. A compass-needle, brought within the influence of such a cir- 
cuit, will arrange itself at right angles to the direction of the flow, and 
under the same circumstances the same end of the needle will always 
point in the same direction. All this is the result of observation and ex- 
periment. 

§ 263. The earth's crust is one vast thermo-electrical circuit, and its 
source of heat is the sun. 

§ 264. In the diurnal motion of the earth, the different portions of its 
tropical regions are heated in succession by the sun during the day, and 
cooled by radiation during the succeeding night. The hotter portions will 
therefore lie to the east and the colder to the west of the sun's place. A 
perpetual flow of electricity is thus developed and maintained in and 
about the earth's crust from east to west, and gives rise to the earth's 
magnetic action. 

§ 265. Were the materials of the earth all equally good electrical con- 
ductors, and the sun always in the equinoctial, the electrical flow would be 
parallel to that great circle, and the compass-needle would always point 
directly north and south. But neither of these conditions obtains. The 
materials vary greatly in conducting power, and the sun's declination is 
ever changing. 

§ 266. The disparity of conducting power directs the electrical flow in 
paths of double curvature, of which the general direction is parallel to 
the equator, and the varying declinations of the sun are perpetually shift- 
ing their precise location and shape as well as changing the intensity of 
the flow. 

§ 267. The position of stable equilibrium, assumed by a magnetic nee- 
dle reduced to its axis, freely suspended from its centre of gravity, and sub- 
jected alone to the directive action of the earth's magnetism, is called the 
magnetic position of the place. 

§ 268. The intersection by a vertical plane through the magnetic posi- 
tion with the celestial sphere, is called the magnetic meridian. 

§ 269. The angle made by the magnetic and the true meridian is 
called the magnetic declination, or simply declination. 

§ 270. The inclination of the magnetic position to the hcrizon is called 
the magnetic inclination or dip. 

§ 271. The magnetic position at the same place is continually varying 
It describes daily a conical surface, of which the place is the vertex, and 



64 



SPHERICAL ASTRONOMY. 



daily mean position the axis, while this axis itself describes a similar sur- 
face once a year about an annual mean position. 

§ 272. The mean of all the declinations and of dips throughout any 
one day are the declination and dip for that day, and are called the 
diurnal declination and dip. The mean of all the diurnal decollations 
and dips for the different days throughout any given year, are the decli- 
nation and dip for that year, and are called the annual declination 
and dip. 

§ 273. The daily and annual fluctuations here referred to are called 
periodic changes. The annual declinition and dip also change, and these 
changes, which are found to take place in the same direction for a great 
many years, are called secular changes. 

§ 274. The magnetic declination and dip vary, in general, with the 
locality. The line connecting those places where the declination is zero, 
is called the line of no declination ; and the line through the places where 
the dip is zero, is called the magnetic equator. 




§ 275. According to the Magnetic Atlas of Hansteen, constructed for 
1787, the line of no declination is found on the parallel of 60° north, a 
little to the west of Hudson's Bay ; it proceeds in a southeasterly direc- 
tion, through British America, the northwestern lakes, the United States, 
and enters the Atlantic Ocean near Chesapeake Bay, passes near the An- 
tilles and Cape St. Roque, and continues on through the southern x\tlantic 
till it cuts the meridian of Greenwich in south latitude G5°. It reappears 
in latitude 60° south, below New Holland, crosses that island through its 
centre, runs up through the Indian Archipelago with a double sinuosity, 
and crosses the equator three times — first to the east of Borneo, then be- 
tween Sumatra and Borneo, and again south of Ceylon, From which it 
passes to the east through the Yellow Sea. It then stretches across the 



TERRESTRIAL MAGNETISM. 65 

coast of China, making a semicircular sweep to the west till it reaches 
the parallel of 71° north, when it descends again to the south, and re- 
turns northward with a great semicircular bend, which terminates in the 
White Sea, 

On the magnetic chart this line is accompanied through all its windings 
by other lines upon which the declination is 5°, 10°, 15°, &c. ; the latter 
becoming more irregular as they recede from the line of no declination. 
The use of these lines is to point out to navigators sailing by compass, the 
bearing of the true meridian from the magnetic. 

§ 276. On the east of the American and west of the Asiatic branch of 
the line of no declination, the declination is west, while to the west of the 
American and east of the Asiatic branch the declination is east. 

§ 277. The magnetic equator cuts the terrestrial equator, according to 
Hansteen, in four, and to Morlet in two points, called nodes, one of which 
is in the centre of Africa. 

§ 278. Beginning at the African node the magnetic equator advances 
rapidly to the north, and quits Africa a little south of Gape Guardafui, and 
attains its greatest north latitude, 12°, in 62° of east longitude from Green- 
wich. Between this meridian and 174° east, the magnetic is constantly 
to the north of the terrestrial equator. It cuts the Indian peninsula a 
little to the north of Cape Comorin, traverses the Gulf of Bengal, making 
a slight advance to the terrestrial equator, from which it is only 8° distant 
at its entrance into the Gulf of Siam. It here turns again a little to the 
north, almost touches the north point of Borneo, traverses the straits be- 
tween the Philippines and the isle of Mindanao, and on the meridian of 
Naigion it again reaches the north latitude of 9°. From this point it 
traverses the archipelago of the Caroline Islands, and descends rapidly to 
the terrestrial equator, which it cuts, according to Morlet in 174°, and 
according to Hansteen in 187° east longitude. Its next point of contact 
with the equator is in west longitude 120°. Here, according to Morlet, it 
does not pass into the northern hemisphere, but bends again to the south, 
while Hansteen makes it cross to the north, and continue there for a dis- 
tance of 15°, of longitude, and then return southward and enter the south- 
ern hemisphere in longitude 108° west, or 23° from the west coast of 
America. Between this point and its intersection with the terrestrial 
equator in Africa, the magnetic equator lies wholly in the southern hemi- 
sphere, its greatest southern latitude being about 25°. 

§ 279. The dip increases as the needle recedes on either side from the 
magnetic equator, the end of the needle which was uppermost in the 
northern being lowermost in the southern hemisphere. 



66 



SPHERICAL ASTRONOMY. 



§ 280. The points at which the magnetic needle is vertical are called 
the magnetic poles. Of these there are four, two in each hemisphere, 
their positions being indicated on the magnetic charts. 

§ 281. On the magnetic charts, the magnetic equator is accom 
panied by curves of equal jp as in the case of the lines of equal decli- 
nation. 

§ 282. The line of no declination and the nodes of the magnetic equa- 
tor are found to have a slow westerly motion, thus causing the differ- 
ent lines of equal declination and dip to pass successively through the 
same place, and illustrating the utter worthlessness of all maps constructed 
from compass bearings unless the diurnal declinations of the needle are 
carefully ascertained and recorded thereon. 

§ 283. The intensity of the earth's magnetic action increases with the 
proximity of the electrical paths to the needle and with the difference of 
temperature in their different parts ; and from changes in these, produced 
by the varying zenith distance of the sun during the day, and of his me- 
ridian zenith distance throughout the year, arise the daily and annual 
mutations of declination and dip ; while to chaDges of the earth's crust, 
produced by geological causes, and increased cultivation of the soil from 
the spread of civilization, are to be attributed the secular variations of the 
same elements. 



TIDES. 

§ 284. Those periodical elevations and de- 
pressions of the ocean by which its waters are 
made to flow back and forth through the 
estuaries that indent our coasts, are called 
Tides. 

§ '285. Perpetual change in the weight of 
the waters of the ocean, due to the attraction 
of the sun and i^oon upon the earth, and the 
diurnal rotation of the latter about its axis, 
cause and mairjtvn the tides. 

§ 286. Let ACJBD be a great circle of 
the earth, in a plane through the sun's centre 
at S. Draw S E through the earth's centre 
at E, and CD through the same point, and at 
right angles to S E. Assume any unit of 
as that at G ; join G and >S, and make 



Fig. 58. 




TIDES, 07 

d = S E = distance of sun from the earth ; 
p = E G = radius of the earth ; 
z = S G = distance of G from sun ; 
<p = AE G = angular distance of G from sun ; 
6 = G S E = angle at sun subtended by radius p ; 
m = mass of sun ; 
k = the attraction of unit of mass at unit's distance. 

Then, since the attraction on unit of mass is proportional to the attract* 
mg mass directly, and the square of the distance inverselj 7 , the sun's action 
on G will be 

km 

©r because 

& ■= d* -+- p 8 — 2 d p cos <p, 

km. 



d 2 + p — 2 d p cos 9 

But each unit of the earth's mass is acted upon by a centrifugal force 
-equal and contrary to the centripetal force impressed upon the unit of 
mass to deflect it from its tangential into its orbital path. This latter is, 
\>y making p = 0, in the above 

km 

and applying this to G in the direction G II parallel to S E, we have all 
the action on G arising from the sun's attraction. 

Resolving these forces into their components in the direction of the 
radius E G, and perpendicular thereto ; also making 

v = resultant of the components in direction of the radius, 
r = "" u " " of tangent, 

and regarding the components which act towards the centre as positive 
and the contrary negative ; also the tangential components which act in 
the direction A G C B A as positive and the contrary negative, we have 

v = -jT cos 9 "~ \n ' . « TT~ ^cos(j)-M) . . (<S) 

a* a -f- p- — 2 a p cos p v 

km . km . 

T = — =- sm q> = 5-= ; . sin (q> + d) • • (v9> 

<P y d 2 -f-p 8 - 2</pcos<p vr ^ ; v 



68 SPHERICAL ASTRONOMY. 

Developing the last factor in equation ( 78 ) T making cos A = 1, because of 
the small value of 0, we have, after reducing, 

2km. p p km . 

v = .(cos 2 — ^.cos^)-| .sm</ ,sm9 

*( 1+ 35- 2 | e<,S *) *(l+S-»|«»#) 

but from the triangle E G #, we have 

^^p.smfr + fl) 
a 

or neglecting 6 in the second member 

. . p . sin <p 
sm d = !- — — -. 
a 

Substituting this and omitting all the terms into which -^ enters as a 

a 

•factor, which we may do without materially altering the value of v, we find 

2 km . p „ km p . „ . . 

v = - r — . cos 2 9 -\ -~- . sm 2 9 . . . . (80) 

Again, omitting d, in the last factor of equation (79% reducing to a 

common denominator and neglecting the terms of which ~ is a factor, we 

have, after replacing cos 9 . sin 9 by J sin 2 9, 

&m p . . . 

t = --J- . sm 2 9 ...... (81) 

§ 287. Making 9 = 0° and 9 = 180° in equation (80), we have the 
effect on the waters at A and at B ; and in both cases 

2km . p 

'= ¥— 

Again, making 9 =a 90° and 9 = 270°, we have the effect on the wa- 
ters at C and D ; and in both cases 

km 

The values of v at ^1 and J5 being negative and those at C and 2? posi- 
tive, and these being connected by a law of continuity, through equation 
(80), the effect of the sun's attraction is to increase the weight of the 
unit of mass, or, what is the same thing, the specific gravities of all bodies 
gradually, in both directions, from A to C and -D, and to diminish them 



TIDES. 



69 




in like manner from C and D to B. 

And tliis being true of all sections of 
the earth through its centre and the 
sun, the waters of the ocean on and 
near the circumference of a section 
through tii 3 earth's centre, and perpen- 
dicular to these, will, by the principles 
of hydraulics, press up those about A 
and B till their increased height shall 
compensate for their diminished speci- 
fic gravity, or till the weights of the 
balancing columns become equal; so 

that the ocean surface will tend to assume, as its form of equilibrium, that 
of an oblongated ellipsoid, of which the longer axis is directed towards 
the sun. The difference of the longer and shorter semi-axes of this ellip- 
soid is about 23 inches. 

§ 288. If the earth had no diurnal rotation about its axis, this ellipsoid 
of equilibrium would be formed, and all would be permanent But the 
earth's diurnal and orbital motion, together with the inertia of water, leave 
no sufficient time for this spheroid to be fully formed. Before the waters 
can take their level, these motions carry the line connecting the earth and 
sun westwardly, and the place of the vertex of the spheroid of equilibrium 
in the same direction, thus leaving that of the actual spheroid to the east 
of the sun, and forcing the ocean to be ever seeking a new bearing. The 
effect is to produce an immensely broad and excessively flat wave, which 
follows or endeavors to follow the apparent diurnal motion of the sun, and 
completes an entire circuit of the earth onee in twenty-four solar hours, 
thus producing a rise and fall of the ocean level twice within this period on 
every meridian. 

§ 289. The rising water is called the food, the falling the ebb tides, and 
the general swell of the ocean is called the primitive tide- wave. 

§ 290. In the open ocean, where the water is deep, and therefore per- 
mits the free transmission of pressure from one remote point to another, 
the motion is one of oscillation in a vertical direction principally. But 
where the tide-wave approaches shoals, such as those along the coasts and 
the beds of estuaries, which intercept the free transmission of pressure, the 
water becomes piled up, as it were, on the side of the open ocean, without 
being able to press up any thing to its support on the land side. It there- 
fore flows inland, an<? produces what are called derivative flood tides. 
After the apex of the tide- wave has passed onward, and low- water sue- 



70 SPHERICAL ASTRONOMY. 

ceeds, the want of support is transferred tc the side of the ocean, the 
water flows out to sea, and forms what are called derivative ebb tides. 
The lines on the earth's surface connecting these places at which high or 
low water, or any other corresponding phases of the tides, occur simulta- 
neously, are called co tidal fines. 

§ 291. The earth and moon are so near to each other, and so remote 
from the sun, as to cause their mutual attractions greatly to predominate 
over the excess of the sun's attraction for one of them over his attraction 
for the other. They therefore revolve about their common centre of 
gravity, and together move around the sun. The attraction of the moon 
for the earth produces upon the ocean effects similar to those of the sun. 

§ 292. The diminution of weight at A and B and increase at C and 
D vary directly as the attracting masses, and inversely as the cubes of 
their distance, equations (80) and (81), and the effects upon the tide- 
wave must be in the same proportion. The mass of the sun is 355000 x 88 
that of the moon, and he is situated at 400 times the moon's distance* 
Whence the effect of the moon at A and B being 

21c ^ m 
~~dT~ r 
that of the sun will be 

2&pm355000x88^ 
(400) 3 1 3 * 

and dividing the last by the first, we have 

355000 X 88 



(400) a 



= 0.488 ; 



so that the effect of the moon is more than double that of the sun. 

§ 293. The lunar day exceeds the solar on an average about 50 min- 
utes ; the lunar tide must therefore move slower than the solar by about 
12°. 5 in 24 solar hours; and hence they must sometimes conspire and 
sometimes oppose one another. The former occurs when the angular dis- 
tance of the sun from the moon, as seen from the earth, is 0° or 180°, and 
the latter when this distance is 90°. 

This alternate reinforcement and partial destruction of the lunar by the 
solar wave, produce what are called spring and neap tides ; the former 
being their sum, the latter their difference. 

§ 294. The sun and moon, by virtue of the ellipticities- of the terres- 
trial and lunar orbits, are alternately nearer to and further from the earth 
than their mean distances. 



TIDES. 71 

' If the mean distances of the sun and moon be substituted in Eq. (80), 
the corresponding ellipticities of the solar and lunar spheroids will be found 
to be 2 and 5 feet respectively ; so that the average spring tide will be to 
the average neap, as 5 + 2 to 5 — 2, or as 7 to 3. 

Substituting the greatest and least distance of the sun in the same 
equation, the resulting tides are called respectively apogean and perigean 
tides ; and representing the ellipticity of the solar spheroid at the mean 
distance by 20, the corresponding ellipticities become 19 and 21. In like 
manner the ellipticities of the lunar spheroid will be found to vary be- 
tween the limits 43 and 59. Hence, the highest spring tide will be to the 
lowest neap, as 59 4- 21 is to 43 — 21, or as 10 to 2,8. 

•§ 295. The sun and moon act to form the apexes of their respective 
tide-waves at different places, depending upon their angular distances 
apart. This gives rise to a resultant wave, whose apex is at some inter- 
mediate place, and the actual tide dag, or interval between the occurrences 
of two consecutive maxima of the resultant wave at the same place, will 
vary as the component waves approach to or recede from one another. 
This variation from uniformity in the length of the tide day is called the 
priming or lagging of the tides — the former indicating an acceleration and 
the latter a retardation of the recurrence of high-water at the same place. 
The priming and lagging are particularly noticeable about the time the 
angular distance between the moon and sun is 0° or 180°, that is, as we 
shall presently see, about new or full moon. 

§ 296. The effort of the attracting body being to form the nearest ver- 
tex of its aqueous spheroid immediately under it, the summit of the lunar 
and solar tide-waves follow the course of the moon and sun to the north 
and south of the equator, and this gives rise to a monthly and annual 
variation in the heights of the principal tides at a given place. 

§ 297. But of all causes of difference in the heights of tides, local 
situation is the most influential. In some places, the tide-wave rushing up 
narrow channels becomes so compressed laterally as to be elevated to extra- 
ordinary heights. At Annapolis, in the Bay of Fundy, it is said to rise 
120 feet. 

§ 298. Were the waters of the ocean free from obstructions due to 
viscosity, friction, narrowness of channels leading to different ports, and 
the like, the time of high-water at a given place, would depend only upon 
the relative positions of the sun and moon, and their meridian passages. 
But all these causes tend to vary this time, and to postpone it unequally at 
different ports. This deviation of the time of actual from that of theoret- 
ical high-water at any place, is called the establishment of the port, and is 



72 SPHERICAL ASTRONOMY. 

ail element of the highest maritime importance. "When ascertained from 
observation, it enables the mariner to know by simply noticing the places 
of the sun and moon with reference to the meridian, when ne may safely 
attempt the entrance of a port obstructed by shoals. 

§ 299. In bays, rivers, and sounds, where tides arise from an actual 
flow of water, the time of " Slack water" or stagnation, must not be con- 
founded with that of high and low water. They may, indeed, coincide, 
but not of course. A river current, for instance, and another from the sea, 
iu; . neutralize each other's flow, while both conspire to elevate the water 
surface ; so, also, an ebbing current may continue its onward course after 
the more advanced part of a returning flood has put its surface on the rise 
by checking its velocity. The same of two currents meeting in a sound. 

§ 300. Starting from A as an origin (Fig. 58), and proceeding in the 
direction of A C B D A, we find the value of t, Eq. (81), negative in 
the 1st and 3d quadrants, and positive in the 2d and 4th ; so that th*» 
tangential components of the solar and lunar attractions conspire with the 
normal to increase the height of the great tide-waves by impressing upon 
the water a motion of translation towards their apexes. But before the 
inertia of the water will permit the latter to acquire much velocity, the 
rotary motion of the earth reverses the direction of the impelling forces, 
and the final effect due to this cause is, in consequence, but small. 



TWILIGHT. 

§ 301. The curve along which a conical surface, tangent to the sun and 
earth, is in contact with the latter body, is called the circle of illumination. 
It divides the dark from the enlightened portion of the earth's surface, and 
is ever shifting its place by the diurnal motion. 

§ 302. The base of the earth's shadow, into which a spectator enters at 
sunset, and from which he emerges at sunrise, is inclosed by an atmospheric 
wall-like ring, illuminated by the direct light from the sun, immediately 
exterior to that which just grazes the earth's surface. The light is reflected 
from the particles of this ring into the shadow, and gives to the air about 
its boundary a secondary and partial illumination called Twilight. A co- 
nical surface through the summit of this ring, and tangent to the earth, 
determines, by its contact with the latter, a limit within which the twilight 
cannot sensibly enter, and twilight will only continue while the spectator 
is carried by the earth's diurnal motion across the zone of which this line 
is the inner, and the circle of illumination the exterior boundary. The 



TWILIGHT. 



73 




belt of the earth's surface over which twilight is visible, is called the cre- 
puscular zone. 

Thus, let EOO'E' be a section Fi & 60 - 

of the earth's surface on the opposite i 

side from the sun ; TAA' T of the ^r 

atmosphere by the same plane, the 
height of the air being exaggerated 
to avoid confusing the figure; and 
SA and S' A' two solar rays tan- 
gent to the earth's surface. The 
particles of air in EA T and E'A' T 
will be illuminated, while those in 
the space EAA'E' will be in the 
shadow. The section will cut from 
the tangent cone the elements A V 
and A' V, which touch the earth at 
and 0\ respectively, and being 
revolved about the line connecting 

the centres of the earth and sun, the part EA T will generate the lumin- 
ous atmospheric inclosure and the points E and 0, the circle of illumina- 
tion and interior boundary of the crepuscular zone, respectively. 

§ 303. To a spectator within the crepuscular zone a portion only of the 
illuminating ring will be visible, and will appear as a blight elliptical seg- 
ment, with its chord in the horizon, its vertex in the vertical circle through 
the sun, and its outline almost lost in the gradual decay of light produced 
by the diffusive action of the air and the progressive thinning and conse- 
quent diminution in the number of reflecting particles towards the summit 
of the luminous ring. 

§ 304. When the spectator is carried obliquely through the crepuscular 
zone without crossing its smaller base, twilight will last all night. 

§ 305. Resuming Eq. (74), that is 

cos z = sin I sin d + cos I cos d cos P ; 

substituting the latitude of the place for I, the declination of the sun for d, 
and the value of P, obtained by converting the observed time from noon 
to the end of twilight in the evening, or from the beginning of twilight in 
the morning till noon, into degrees, the average value of a number of de- 
terminations for z will be found to be about 108°; so that at the end of 
evening or beginning of morning twilight the sun is 18° below the hor'zon. 
§ 306. From the above equation we find 



74 SPHERICAL ASTRONOMY. 

. cos z — cos / . cos d . cos P 

sm I = 



sin c/ 

The angle P S Z, made by the hour 
circle P £ and vertical circle Z S, is 
called the variation or the parallactic 
angle. Denote this by |, then from the 
triangle Z P S, will 



(i 



. sin Z = sin <Z cos s + cos d sin s cos £. 




Equating the second members of this 
and the equation above, we have 



(2) . . . . cos I . cos P == cos z . cos d — sin z sin cZ . cos if ; 
and if the sun be in the horizon, then will 

z — 90°, P = P', and £ = £', and 

(3) . . . . cos I . cos P' = — sin d . cos £'. 
Also, from the same triangle, 

(4) . . . . cos Z . sin P = sin . sin £; 
and when the sun is in the horizon, 

(5) . . . . cos I . sin P' = sin £'. 

Multiply (2) by (3), also (4) by (5), and add the products, there 
result, 



wilj 



cos 2 I . cos (P— P) = — cos z cos d sin d cos £' + sin z cos (£ — £') — cos 2 rf sin s cos £ cos £'. 

From (1), we have 

sin I — sin d . cos z 



cos £ = 



cos d . sin z 



and for the sun in the hori; 



cos £' = 



sin I 

cos d ' 



82) 



(83) 



TWILIGHT. 75 

which substituted above, give 

cos 2 I . cos (P — P') = sin z . cos (£ — I') — sin 2 I; 
whence, because 

cos (P - P') = 1 - 2 sin 2 1 (P - P'), 



we have 



sin' I (P - />') = 1 - sin , . co s (g^T) * 

• 2 v ' 2 cos 2 / 



passing to the arc and making 

t 
we have 



/ l - sin g . cos (g - g) 

V ^o^T ' ' (84) 



2 . 
t = — sin 

15 V 2 cos 2 I 



which will give the time required for the sun, or other heavenly body, 
to pass from the horizon to a zenith distance z, or, conversely, from a 
zenith distance z to the horizon. 

Making z = 90° + 18° = 108°, Eq. (84) becomes 



t = A. rin-.y/ 1 -™ 'f •?"(*'-*•). . . (85) 

15 V 2 cos 2 / v ' 

which will give the duration of twilight for any latitude and season of 
the year ; and for this purpose, the values of £ and £' must be found 
from Eqs. (82) and (83), after making, in the former, z = 90° + 18°. 

The value of t, in Eq. (85), becomes a minimum when j = £', and 
for the duration of the shortest twilight, we have, after replacing 
1 — cos 18° by its equal 2 sin 2 9°, 

2 

t = -^ . sin" 1 (sin 9° . sec I) (86) 

15 

Equating the second members of Eqs. (82) and (83) 

sin d = — tan 9° . sin I (87) 

In a given latitude, Eq. (86) will make known the shortest twilight, 
and Eq. (8*7) the season at which it will occur. 

* Ann Arbor Astronomical Notices, N . 1. 



76 



SPHERICAL ASTRONOMY. 



§ 308. The sign of the second member 
of Eq. (87) shows that at the time of 
shortest twilight the spectator and the sun 
will be on opposite sides of the plane of 
the equinoctial. 

§ 309. The depression of the lowest 
point Q' of the equinoctial below the ho- 
rizon HH\ is 90° — l\ and of the low- 
est point S of the sun's diurnal path, 
when his declination is of the same name 
as the spectator's latitude, 90°— (I + d) ; 
and when 

90° -ll + d) = 18°, 

the end of the evening will be the beginning of morning twilight, and the 
nocturnal path of the spectator will be tangent to the inner boundary of 
the crepuscular zone. 







/ 



THE SUN. 



§ 310. The Sun, as before stated, is the central body of the solar sys- 
tem, and fi'om this circumstance gives to the latter its name. It occupies 
one of the foci of all the elliptical orbits of the planets, and, of course, that 
of the earth. 

§ 311. Distance and Dimensions of the Sun. — Its horizontal parallax 
denoted by P, and apparent semi-diameter denoted by s, vary inversely 
as the earth's radius vector. For the mean radius it is found, § 113-6, 

P = 8".6, and s = 16' 01 ".5 ; 

which in Eqs. (28) and (29) give 



r n = P ' p 



206264".8 ^ ono , 
p • —^ir- = 23984 • P 



(88) 



d = 



16' 01".5 
8".6 



961".5 
8".6 



= 111.5 p 



(89) 



From Eq. (88) it appears that the mean distance of the earth from the 
sun is 23984 times the earth's equatorial radius; and from Eq. (89) that 
the sun's diameter is 111.5 times that of the earth. The volumes of these 
bodies are as the cubes of their diameters, and hence the volume of the. 
sun is 1384472 times that of the earth. 



THE SUN. 77 

§ 312. If the equatorial radius p be replaced in Eqs. (88) and (89) 
by its value in miles, § 98, we find 

r„ = 95,043,800 miles, 
2tf= 882,000 " ; 

that is to sa)' - , the mean distance of the earth from the sun is, in round 

numbers, about 95 millions of miles, and the diameter of the sun is 882 

thousand miles. The mean distance of the earth from the sun is assumed 

as the unit of linear dimensions in all celestial measurements. 

*— § 313. Mass of Sun. — In Analytical Mechanics, § 201, we find the 

equation 

'-=VT' < 8 '" 

in which T denotes the periodic time of a body revolving about a centre 
of attraction, a the mean distance of the body from the centre, ir the ratio 
of the circumference to the diameter, and k the attraction on a unit of 
mass at the unit's distance. 

Let k become ^ in the case of the sun's action on the earth ; then will 
T become the sidereal year, and a the semi-transverse axis of the earth's 
orbit, and 

^-^T- (90) 

and for the action of the earth upon the moon 

p'**-jnr- ( 91 ) 

in which /a/ denotes the attraction on the unit of mass at the unit's distance 
exerted by the earth. 

Now the attractions exerted by two bodies on the same mass at the 
same distance, are directly proportional to their masses respectively ; and 
denoting the mass of the sun by M, and that of the earth by M' we have 

M j* T" a* 
f* 
But in Eqs. (62) and (88) 

T = 365 d .25, and a as 23984 . p ; 

and we shall presently see that the moon revolves about the earth once in 
27.5 days, at a mean distance of 60 times the equatorial radius of the 
earth * Making, therefore, 



It " 11/ ~ T % ' a' 3 (92) 



78 SPHERICAL ASTRONOMY. 

T = 2Y.5, and a' = 60 . p, 

and substituting above, we have 

M 

— = 354936. 

M 

That is, the sun contains 354,936 times as much matter as the earth; and 
as the common centre of inertia divides the line joining their respective 
centres of inertia into two parts, which are inversely proportional to their 
masses, the common centre of inertia of the sun and earth, about which 
both bodies would describe their respective orbits were they undisturbed 
by the other bodies of the system, is but 267 miles from the sun's centre, 
or about 3^0 o tn P art °^ ^ s own diameter. 

§ 314. Denote by D the density of the sun, and by Fits volume; also 
by D' and V\ respectively, the density and volume of the earth ; then, 
Analytical Mechanics, § 18, 



and by division 



and substituting the ratio of the masses and of the volumes given above, 
we find 

D = 0.2543 .D'\ 

so that the sun is but a trifle more than one-quarter as dense as the earth. 
The latter is known, from the recent experiments of Mr. Francis Baily, to 
be 5.67, the density of water being unity; and this value substituted for 
D' above, makes the density of the sun not quite once and a half that of 
water. 

§ 315. Surface Gravitation of the Sun. — By the laws of gravitation, 
the attraction of one body upon another varies as the quantity of matter in 
the attracting body directly, and the square of the distance through which 
the attraction is exerted, inversely. The distance is that between the cen- 
tres of gravity of the bodies. 

Denote by W and W the weights of the same body on the surfaces of 
the sun and earth, respectively ; then will 

whence *=£..£.. (93) 



M = 


D 


.v, 


M' = 


D' 


V'\ 


M 


D 


. V 
V 



THE SUN. 



70 



= 28,5. 



and substituting the values just found, 

W 

W 

That is, a body weighing one pound at the equator of the earth \sculd 
weigh 28,5. pounds at that of the sun ; and acquire, therefore, during each 
second of its fall a velocity of 916,44 feet. 

§ 316. Sun's Hotation and Axis. — Through the telescope 
the sun's surface often exhibits dark spots which slowly 
change their places and figure. They cross the solar disk 
from east to west, and thus reveal a rotary motion of the 
sun itself from west to east about an axis. 

§ 317. To find the time of rotation and the position of 
the axis, it will be necessary first to find the heliocentric 
longitudes and latitudes of the same spot at different times. 
To do this, let S be the sun's centre, E that of the earth, 
P the spot, and N its projection upon the plane of the 
ecliptic. Make 

I = heliocentric longitude of the earth ; 

x = " " " spot; 

y = P SN = heliocentric latitude of spot ; 
8 = P EN = geocentric latitude of spot ; 

e = SEN = difference of geocentric longitude of the sun and the spot, 
J = sun's apparent semi-diameter. 




o t 



Then 



SP sin y = PN= EP sin (3 = SE sin /3, 



because the difference between EP and S E is insignificant in comparison 
with either ; whence 

SE .' sin/3 



Again 
■whence 



sin y = -q-b • sm p = -r—z 

of sin 4 



SP . cos y : EP . cos j8 : : SN : NE, 

: : sin e : sin (I — x) \ 



(94) 



•HX 



sin (I — x) = 



and replacing cos y by its value, 
sin (I — x) = 



sin 


e . 


cos 





EP 




cos 


y 




SP 1 


sin e . 


cos 


/B 




sin 


^ . 


cos 


y 






sic 


€ . 


cos 


/3 






V sin 2 J — sin 2 /3 




80 SPHERICAL ASTRONOMY. 

or for logarithmic computation, 

. ,-, v sin e . cos ft . „. 

sin (/ — «)= . . (9o) 

-y/sin (<4 + £) • sin (^ - 8) 

§ 318. Position of the Suns equator, and the time of the Surfs rota- 
tion. Let E be the pole of the ecliptic, P that of the sun's equator; A, 
A , and A" the heliocentric places of the 
same spot observed at three different times ; 
and let E A, EA, EA", PA, PA', PA" 
be the arcs of great circles. The first three 
are known from Eq. (94), being the helio- 
centric colatitudes of the spot; as also the 
angles AEA, AEA", and AEA" from 
Eq. (95), being the differences of the he- 
liocentric longitudes — all deduced from ge- 

osurface observations of the spot's right ascension and declination, § 152. 
All the sides and angles of the triangles AEA', AEA", and A' EA" 
may be found, two sides and the included angle in each being given ; 
hence the sides A A', A' A", and A" A, and the angles A, A', and A", in 
the triangle A A' A", are known. Now P being the pole of the sun's 
equator, parallel to which the spot revolves, 

PA = PA'= PA") 

Make 

2S = A + A'+A" = 2PAR + 2PAA + 2PAA" 
= 2PAR + 2A f : 
whence 

PAR = S- A, 

and PAR becomes known. 

If P R be perpendicular to A A", 

AR = \AA"\ 

then in the right-angled triangle APR, the angle at A and the side A R 
being known, the side PA is computed; and, finally, in the triangle 
APE, the sides A P and A E, and the angle EA P^EAA" -PA A" 
being known, P E is computed. 

§ 319. The arc EP is the heliocentric colatitude of the pole of the 
sun's equator, and the angle AEP, added to the heliocentric longitude of 
the spot at A, gives its heliocentric longitude. The position of the sun's 
equator becomes, therefore, known. The heliocentric latitude and longi- 
tude of its north pole at the beginning of the present century were, respec- 
tively, 82° 30' and 350° 21'. 



THE SUN". 



81 



Fig. 65. 




From the triangle APR the angle APR becomes known, the double 

of which is A PA". Then, denoting by T the time of one rotation, and 

by t the interval between the observations on the spot at A and A", we 

have 

APA" : t :: 360° : T; 

whence T is known to be about 25.325 days, making the angular velocity 
of the sun around its axis about one twenty-fifth that of the earth. 

From this motion it is concluded that the sun is flattened at its poles. 

§ 320. Physical constitution of Sun. — 
The study of the solar spots has led to inter- 
esting conclusions in regard to the physical 
constitution of the sun itself. The spots are 
transient in character, variable in size, shape 
and number, and confined to two compara- 
tively narrow zones parallel to, and at no 
great distance from the sun's equator. They 
appear perfectly black, and surrounded by a 
border less dark, called a penumbra. The 
black part and penumbra are distinctly de- 
fined in outline, and do not fade the one into the other. 
Sometimes this penumbra presents two or more shades, and 
in this case also there is no gradation, but well-marked out- 
line, indicating a total absence of blending. 

As the spots move towards the edge Fi s- 67 - 

of the sun, the penumbra on the inner .-■.-■My'- 

side gradually contracts, and with the 
black spot disappears before reaching 
the boundary of the disk ; the penum- 
bra on the outer side expands, and is 
the last visible remnant of the spot as it passes behind the sun. At its 
reappearance on the opposite edge of the sun, the spot exhibits similar 
phenomena — the penumbra first appears, then the black portion on its in- 
ner side, the contraction of the penumbra in width, and its extension 
around the black till the latter is entirely surrounded. 

This is precisely the appearauce that would be presented by a deep pit 
or excavation with a dark or non-luminous bottom. The rotation of the 
sun would bring the slanting surface leading from the inner edo-e of its 
mouth more and more in the direction of the spectator till it would be lost 
in the foreshortening, the inner edge would presently mask the bottom, 
and the surface of the opposite side would be turned so nearly perpendicu- 








SPHERICAL ASTRONOMY 



larly to the line of sight as to appear broadest just before passing behind, 
at disappearance, or at reappearance, to the front of the sun. 

§ 321. The spots gradually expand or contract, change their figure, 
vanish, and break out again at new places where none were before. When 









disappearing, the central black part contracts to a point and vanishes be 
fore the penumbra ; and a single spot is sometimes seen to break up into 
two or more smaller ones. 

§ 322. A circle of which the diameter is one second is the smallest vis- 
ible area. A single second at the earth is subtended at the sun by a dis 
tance of 461 miles, and the area of the least visible circle on the sun's 
surface is, therefore, 167,000 square miles. A spot whose diameter was 
45,000 miles has been known to close up and disappear in course of six 
weeks, thus causing the edges to approach one another at the rate of 1000 
miles a day. Many spots distinctly visible have been observed to vanish 
in a few hours, indicating a degree of mobility inconsistent with the idea 
of solids and liquids. 

§ 323. Light proceeding very obliquely from the surfaces of incandes- 
cent solids and liquids is always polarized, whereas that from gases under 
the same circumstances is not. The light from the edge of the solar disk 



THE SUN. 83 

leaves the surface of the sun in a direction nearly coincident with the 
surface itself, and yet when examined by the usual tests exhibits no signs 
of polarization, 

§ 324-. The luminous part of the sun is not uniformly bright, but pre- 
sents a moLtled appearance, and immediately about the spots are often 
seen well-defined and branching streaks, called facules, brighter than 
other paits of the surface; among these, spots often make their appear* 
auce. They are best seen near the border of the disk, 

§ 325, The brightness of the solar disk sensibly diminishes towards 
the borders; and this fact has given rise to the supposition that the sun 
is surrounded by an atmosphere not perfectly transparent, and of great 
extent above the luminous envelope. The loss of light towards the bor- 
ders would result from the greater absorption of the luminiferous waves 
in consequence of traversing a greater thickness of the atmosphere in 
that direction. 

§ 326. The moon, of which an account will be given presently, is 
known to be a non-luminous, opaque, spherical mass, and so near the 
earth as to give to it an apparent diameter about equal to that of the 
sun. This little body often interposes itself so as completely to conceal 
the sun from view, producing what is called a solar eclipse. At the in- 
stant of greatest solar obscuration — that is, when the moon completely 
covers the sun — red protuberances resembling flames of fire are seen to 
issue apparently from the edge of the moon, but in fact from that of the 
sun, revealing the existence of intense commotion and physical changes 
about the surface of the latter body. 

§ 327. From all which it is inferred that the sun is an opaque solid, cov- 
ered by a gaseous envelope of well-defined boundary and intense luminosity, 
the whole being surrounded by a non-luminous atmosphere of vast extent 

No explanation free from objection has, thus far, been given for the 
solar spots. Some have supposed them to arise from scoria or flakes of 
incombustible matter floating upon the sun's surface; while others, with 
perhaps greater reason, have attributed them to temporary openings in 
the photosphere that envelops the sun, exposing to view detached por- 
tions of his solid crust, which appear black from contrast. 

But it must not be inferred from this that the solid portion of the sun 
is regarded as non-luminous. Were he stripped of his gaseous coating, 
he would no doubt shine with diminished but yet intense brilliancy. A 
piece of quicklime, in a state of most active combustion under the action 
of a compound blowpipe, is, when projected upon the bright part of the 
sun, as dark as the darkest part of the spots. 

During the interposition of the lunar screen between the sun and a 



84 SPHERICAL ASTRONOMY. 

spectator on the earth, the surrounding landscape takes on the ob&'irte 
illumiuation produced by a closing evening twilight, and the tempera tire 
is always sensibly depressed, thus corroborating the suggestions of other 
phenomena, that the sun is the great source of light and heat to the i-artii. 
But light and heat are the results of molecular agitation. What, 
then, is the cause of that perpetual molecular vibration essential to the 
self-luminosity of the sun ? The solar system is believed to have resulted 
from the subsidence of a vast nebula ; the planets and satellites are de- 
tached fragments left behind in the progress of the general mass towards 
the centre ; the sun itself is the central accumulation. This nebula 
must have extended originally far beyond the orbit of Neptune, the ex- 
terior planet now known. The distance of this planet from the sun is 
more than thirty times that of the earth. The condensation has taken 
place under the action of weight impressed upon the elements by their 
reciprocal attractions for one another. The living force with which so 
much matter would reach the terminus of a fall necessary to transfer it 
to its present abode, could not fail to impress upon the condensed mass 
the most intense molecular agitation. This agitation, or molecular liv- 
ing force, can only be lost through the agency of the surrounding me- 
dium which diffuses it through space ; and the loss in a given time is 
determined by the density of the medium, being less as the density is 
less. The medium which pervades the planetary space is so attenuated 
as to offer no sensible resistance to the denser bodies that move through 
it, nor could we be conscious of its existence at all but for the almost 
inconceivably small amount of living force which it brings from the sun 
to impress upon us the sensations of light and heat. A process so slow 
would require countless ages to bring the solar molecules to rest, and 
convert the sun into a non-luminous mass. 

PLANETS. 

§ 328. Let us now resume the Planets. As before remarked, these 
bodies move in elliptical curves, of which one of the foci of each is at the 
centre of the sun. A spectator on the earth views these bodies, therefore, 
from a station far removed from their centre of motion, and even from the 
planes of their orbits. Hence, their co-ordinates of place, measured by 
the aid of instruments, are affected with both geocentric and heliocentric 
parallaxes. To eliminate these, and then from the resulting heliocentric 
co-ordinates to determine the elements of a conic section whose curve 
shall pass through the observed places and have a focus at the sun's cen- 
tre, is the object of one of the most important problems in Astronomy. 



Plate H. 




TO FROtfT PA^E £# « 



PLANETS. 85 

Three observed right ascensions and declinations, together with the inter- 
vals of time between the observations, are sufficient for its solution. 

§ 329. The planes of the orbits passing through the sun, the orbits 
themselves will pierce the plane of the ecliptic in two points, called nodes. 
The node by which the body passes from the south to the north of the eclip- 
tic is called the ascending node ; the other is called the descending node. 

§ 330. The angle which the plane of a body's orbit makes with that 
o£ the ecliptic or equinoctial, is called the inclination. 

§ 331. The semi-transverse axis, called the mean distance, and eccen- 
tricity, determine the size and shape of the conic section. 

§ 332. The inclination, heliocentric longitude, or light ascension of the 
ascending node, and of the perihelion, fix the position of the orbit in space. 

§ 333. The time of the body's being at perihelion, and its mean angu- 
lar velocity, called its mean motion, give the circumstances of the body's 
motion in the orbit. 

§ 334. The orbit of a heavenly body is therefore completely deter- 
mined when the inclination, mean distance, eccentricity, longitude of the 
ascending node, longitude of the perihelion, epoch of the perihelion passage, 
and mean motion are known. These are called the elements of an orbit. 
They are seven in number. 

§ 335. To find a planefs elements. — The polar equation of the orbit is 

a (1 - e 2 ) 

r = T ± L (96) 

1 4- e cos v 

in which r is the radius vector of the planet, a the semi-transverse axis, 
called the planet's mean distance, e the eccentricity, and i the planet's an- 
gular distance from perihelion, called the true anomaly ; the pole being 
at the sun. 

Making v = 90°, r becomes the sem'i-parameter, which denote by L, 
and we have, Eq. (96), and Analyt. Mechanics, § 200, 

L = a(l-S) = ±f- (97) 

in which c denotes the area described by a radius vector in a unil of time ; 
and Eq. (96) may be written 

r = — (OS) 

1 + e cos v 

whence e cos v = 1 (99) 

r 

from which, denoting the planet's velocity in the direction of the radius 
vector by V n we find, Appen lix VI., 



36 SI IIERICAL ASTRONOMY. 

esinv = — .V r (100) 

which divided b) Eq. (99) gives 

tan ^ = — :-^— .V r (101) 

§ 336. Denote by jo, the perihelion distance, then, making v = 0, in 

iq. (96), 

^ == a (1 - e) (102.) 

§ 331. Denoting by T* the periodic time, we have, An. Mm* § 201, 

^# <™> 

and denoting the mean motion by », 

2* V& . 

a 2 

§ 338. Take an auxiliary angle, such that 

cos u — e . „ % 

cos » = (105) 

1 — e cos u 

then, Appendix VIL, n t = u — e sin « (106) 

in which * denotes the time from perihelion, and », as above, the metm 

motion. 

§ 339. The product n t is the angular distance which the planet would 
be from perihelion had it moved from that point with its mean motion n 7 
and is called the mean anomaly. 

§ 340. The auxiliary angle u is called the eccentric anomaly, and dif- 
fers from n t only because of the eccentricity of the orbit ; for if the latter 
be zero, n t will equal u* 

§ 341. From Eq. (105) we readily find 

u t/l— e v , _* 

•"rV^i ( } 

§ 342. Making in Eq. (103), k = jx, a ■ = 1, and 1"= 365 d .256, we find 
log /j, = 6.4711640 
log VfA =e 8.2355820 
§ 343. From the centre of the sun draw right lines respectively to the 
vernal eq linox, intersection of the solstitial colure with the equinoctial, and 
north ce'estial pole, and take these as the axes ar, y, and z. The planea 
of the equinoctial, of the equinoctial colure, and of the solstitial colure, 
will be the co-ordinate planes xy y xz y and z y respectively. 



PLANETS. 



87 



Denote by V„ F y , and V z the con: ponent velocities of the planet in the 
direction of the axes, and by c', c", and c'" the projections of c on the co- 
ordinate planes x y, z y, and z x respectively ; then, Analytical Mechanics 
§ 189, equations (260), will 

xV y -yV x = 2c', 
y V z -zV y = 2c", 

zV x -xV z = 2c f ", 

and 

c 2 = c 12 + c" 2 -f c'" 2 

§ 344. Denote the inclination of the orbit to the plane of the equinoc- 
tial by i, then will 

("0) 



(111) 

(112) 



(108) 
(109) 





c 


OS l = — 
c 


§ 345. Also, 








r 2 = 


x 2 + f+z'< 


and, Appendix VIIL, 







v r = x -.v x + y -.v y + z -.v z 



§ 346. Let S be the sun, 
P the place of the planet, R 
that of the perihelion, B the 
vernal equinox, E the summer 
solstice, A the north celestial 
pole, BE the ecliptic, R'P'N' 
the intersection of the plane of 
the planet's orbit with the ce- 
lestial sphere, N' the heliocen- 
tric place of the ascending 
node N on the equinoctial, 
A P'P'" and A B f R" quad- 
rants of great circles of the ce- 
lestial sphere. Make 

X = B P"'= the planet's 

heliocentric right as- X 
cension. 

S = A P' = the planet's heliocentric north polar distance. 

r] = N'P'"= distance in heliocentric right ascension from the node. 

f = B N' — heliocentric right ascension of ascending node. 

<p = JV'P' = distance of the planet from the node. 

>/ = N'R" = distance of perihelion in right ascension from the asc. node. 

vt = B R" = heliocentric right ascension of the perihelion. 




88 
Then 



SPHERICAL ASTRONOMY. 

y 



tan X = 



(113) 



tanP"S£=taTiP'CP"' = cot P> A = -; 

x 



and in the triangle A P' C, the 
side ^Cbeins: 90°, 



cot 8 = cos X 



(114) 




Again, in the triangle PP'"N', 
right-angled at P'", 

sin y\ ■= cot S • cot i (115) 

S = X-7] (116) 

tan 9 = sec i- tan (X— s) (117) 

In the triangle N'R'R", right- 
angled at is?", 

tanX'= cos i.tan ((p+v) (118) 
in which v is the true anomaly 
F'SR' ; and hence 

tf = X' + s (119) 

§ 347. It thus appears that as soon as x, y, z, V x , V y and V t are 
found, all the elements become known ; and the preceding formulas, 
arranged in the order of sequence, will stand 



a) 

(2) 
(3) 

(5) 
(6) 
(7) 



r'= z' + f + f; 
2c'.= xr,-yV.; 

2e'"=zV,-xV ! ; 
e'= e' 2 +c" 2 +c"". 

r ic * 

i= F' 

r. = -.r.+ Z.v,+~r.. 



L 

tEJl V = — 



2c L-r 



■V.. 



~.(--l);Eq.(99). 

3S v \r J ' 



PLANETS. 89 

(8) « = ^£*$i Eq (96). 

(9) p = a(l- e ). 

do) » = 4- 

(11). . . . coB^ = ff P + V Eq:(105). 

1+ecosv v y 

(12) , = ?f=™ Eq . (106). 

(13) cos i = — 

€ 

(14) tan X -- y . 

x 

(15) cot £ = cos X. -• 

X 

(16) sin y\ = cot £ . cot t. 

(17) . . . . . s = X-yj. 

(18) tan 9 = sec % . tan (X — s). 

(19) tan X'— cos i. tan((p + v). 

(20) m = X'-f s. 

For the method of finding ar, y, z, V xi V yt V t , see Appendix IX. 

§ 348. The sign of c / in Eq. (110) determines the inclination to be 
acute or obtusp, and also the direction of the motion, the latter being 
direct when c' is positive, and retrograde when negative. The planet 
will be receding from or approaching the equinoctial according as z and 
V z have the same or opposite signs, and it will be north or south of the 
equinoctial according as z is positive or negative. The signs of x and 
y will show in which quadrant the planet is projected on the plane of 
the equinoctial. See Appendix I. 

§ 349. The position of the orbit is given in reference to the equinoc- 
tial; to obtain it in reference to the ecliptic is a mere operation of 
spherical trigonometry too obvious to require explanation. 



90 SPHERICAL ASTRONOMY. 

§ 350. The disturbing action of the planets upon one anothei causes the 
nodes, inclinations, eccentricities, and perihelions to vary. The mean rate 
of change in each case is found by dividing the whole change, as ascer- 
tained at epochs widely separated, by the interval. 

§ 351. The periodic time in mean solar days is found by multiplying 
the tabular periodic time, which is expressed in that of the earth as unity, 
by 365.24 ; and the mean distance in miles will be given by the product 
of the tabular distance into 95,000,000. 

§ 352. Dimensions and Geocentric Distances. — Denote by X, Y, aud Z 
the co-ordinates of the earth ; by x, y, and z, those of a planet, referred to 
the centre of the sun ; and by D the distance of the planet from the earth. 
Then 

D = V(X- xf + ( Y- yf + (fe - zf .... (120) 

§ 353. The horizontal parallax of any body is the apparent semi-diame- 
ter of the earth as seen from the body. Let ie' be the horizontal parallax 
of the sun, P' that of the planet, and r the radius vector of the earth ; 
then, as the apparent semi-diameter of the earth is inversely proportional 
to the distance from which it is viewed, will 



whence 



and this in Eq. (29) gives 



* - P> • • X ■-■-■ 1 



?' = «^ ( 121 ) 



D 



rf = P-4 • • •• -022) 



P> 



in which s is the planet's apparent semi-diameter measured with the mi- 
crometer, d its real semi-diameter, and p the earth's equatorial radius ; 
whence the diameter, surface, and volume of the planet become known. 

§ 354. Mercury and Venus are called inferior planets, being lower 
or nearer to the sun than the earth; the others are called superior 
planets, because they are higher or more distant from the sun th'an the 
earth. 

§ 355. When the geocentric longitude of a body is the same as that of 
the sun, the body is said to be in conjunction; when its longitude differs 
by 180°, in opposition. The superior planets may be in opposition, but 
the inferior planets never. 

§ 356. A body in conjunction or opposition is also said to be in syzygy. 



PLANETS. 



91 



§ 357. When an inferior planet is in perigean syzygy, it is said 
be in inferior conjunction ; when in apogeau syzygy, in superior < 
junction. 

§ 358. Synodic revolution. — The interval Fig 71. 

of time between two consecutive returns of a 
planet to apogean or perigean syzygy is 
called its synodic revolution. 

Denote by m the heliocentric mean daily jg 
motion of the earth in longitude ; by n, that 
of any planet ; and by T, the length of its 
synodic revolution; then will m ~ n be the 
relative motion in longitude of the earth and 
planet, and 






f) 



%:. 






'1 



§ 359. Geocentric Motion in Longitude. — 
The angle at the earth, subtended by a body's 
linear distance from the sun, is called the 
body's elongation ; the projection of a body's 
centre on the plane of the ecliptic, is called 
the reduced place ; and the projection of its 
radius vector, is called the curtate distance. 

Thus, let S be the sun, P a planet, E the 
earth, and P JV a perpendicular from the 
planet to the plane of the ecliptic, intersecting 
the latter in N; then will SEP be the 
elongation, N the reduced place, and S N 
the curtate distance of the planet. 

^§ 360. Draw S V and E V to the vernal 
equinox ; they will be sensibly parallel. Also 
draw N N, and E E t perpendicular to E V 
and S V, and make 



a = S 2V = mean curtate distance ; 

p = EJV= earth's distance from the reduced place 

I = VS JV= planet's heliocentric longitude; 

n = hourly change in the same ; 

L = V S E = earth's heliocentric longitude; 

X be V E N "= planet's geocentric longitude; 

m — hourly change in the same. 



( 



m 



.yf 




92 SPHERICAL ASTRONOMY. 

Then, the mean distance of the earth from the sun being unity, will, 

Appendix X., 

m = P 2 [a 2 + a 2 — (a + a?) . cos (L — l)].n. . . (124) 

in which 

_ cos X 

a cos I — cos L 

and which will make known the rate and direction of the body's motion 
in geocentric longitude. 

§ 361. Direct and Retrograde Motion ; Stations. — When the planet is 
in apogean syzygy, then will L — I = 180°, cos (L — I) = — 1 ; and, 
Eq. (124), 

m = P 2 .a.(a + l) (1 + a*).n (125) 

and m will always be positive ; that is, the geocentric motion of the planet 
will be direct. 

§ 362. When the planet is in perigean syzygy, then will L — I = ; 
cos (L — I) — 1 ; and, Eq. (124), 

m== P*.a(a- 1) (1 —a?).n (126) 

and m will always be negative, whether a be greater or less than unity ; 
that is, the geocentric motion of the planet will be retrograde. 

§ 363. In changing from direct to retrograde, and the converse, the 
body must appear stationary. This will make m = 0, and, Eq. (124), 

cos (L - V) = 3 = 1 (127) 

1 + « 2 a 2 + a~ 2 — 1 

a quantity which is always less than unity, whether a be greater or less 
than unity; that is, all the planets must sometimes appear stationary. 
The condition expressed by Eq. (127), may always be satisfied for two val- 
ues of L — I. The two places of a body, in which it appears stationary, 
are called stations, 

§ 364. Let the value of L — I for one of the stations be 9 ; then, Eq. 

(124), 

3 5. 

= P 2 [a 2 + a 2 — (a + a 2 ) cos 9] . n ; 

and subtracting from Eq. (124), 

5. 

m = P 2 . (a + a 2 ) [cos 9 — cos (L — I)] . n . . . (128) 

in which, as long as <p is less than 90°, and L — I greater than <p and less 
than 360° — 9, m will be positive ar>d the motion direct. 



PLANETS. 



93 






§ 365. Denote by n T the earth's mean 
motion in longitude; then will n ~ n' be 
the mean relative heliocentric motion of 
the earth and planet ; and denoting by t r 
and i d the durations of the retrograde and 
direct motions, we have 



Fig. 7a 



L = 



2<p 



(129) 



L = 



360° — 29 



-— r - . . (130) 




and the duration of the direct motion will be the longer. 

It thus appears that in the course of one synodic revolution the planets 
appear sometimes to be stationary, then to move forward or in the order 
of the signs, then to be stationary again, and finally to move backwards. 

366. Phases of the Planets. — A body illumined by the sun, and 
shifting its place in reference to the sun and earth, presents to the latter 
different appearances at different times. These appearances are called 
phases. 

§ 367. To find the phase F1 s- u - 

of a globular body, let S be /' E 

the place of the sun's centre, 
E that of the earth, and P 
that of the body; ADCB 
a section of the body by a 
plane through E, P, and #; 
a plane through P and per- 
pendicular to P E will cut 
from the body's surface the 
section AMCJV, which de- 
termines the hemisphere 

turned towards the earth ; and another through P, and perpendicular to 
SP, will give the section MDNB, which determines the illuminated 
hemisphere turned towards the sun. The illuminated lower surface 
M CNBM will be visible to the earth, and its projection on the plane 
AM CN will give the shape and magnitude of the phase. The projection 
of the semicircle MB JVM will be a semi-ellipse MB' JVM, of which the 
transverse axis is equal to the diameter of the body ; its conjugate will 
vary with the angle which the projected plane makes with that of projec* 
tion. The phase will therefore have for its boundary a semicircle on the 




94 



SPHERICAL ASTRONOMY. 



side towards the sun and a Fi £- u bis - 

semi-ellipse on the other, / 

these being united at the 

extremities of a common 

diameter. When the phase 

is concave on the elliptical 

side, it is called crescent; 

when con vex, gibbous; when 

straight, dichotomous ; and 

when the ellipse becomes a 

semicircle, full. Make 

d = distance EP\ 

a = apparent area of the semicircle MONM2X distance unity, 

a'= " " " " distanced; 

p= « w phase MCJSFB'M 
e= " " semi-ellipse MB'NM " 
d = angle B P B' = S PE\ the exterior angle of elongation. 




Then 
but 

whence 



e = a' cos #, 



J) = fl' (1 - cos £). 
The apparent diameter of the body vanes inversely as the first, and the 
apparent area of the disk as the second power of the distance ; whence 



/ a 



which substituted above gives 

a , 

' = *<>- 



= 15 . ver sin & 



(131) 



§ 368. The orbits of the principal planets have but slight inclinations 
to the ecliptic. At inferior conjunction of the inferior planets, the exterior 
angle of elongation will therefore approach to 0°, and the distance will be 
the least; at superior conjunction the exterior angle will be 180°, and the 
distance the greatest. In the first position the planet will be invisible, in 
the second full, and between these limits the phase will pass through cres- 
cent, dichotomous, and gibbous, with a continually decreasing diameter. 
From superior to inferior conjunction the same phases occur, but in the 
reverse order. 

In the case of the supe r ior planets, the exterior angle of elongation ap- 



PLANETS. 



95 



proaches to 180° both at conjunction and opposition, and it never can be 
as small as 90°. The phases of these bodies must, therefore, always be 
either gibbous or full ; largest in opposition, and smallest in conjunction. 
If S be the place of the sun ; E that of the earth ; F,, V 2 , &c, the 



Fig. T5. 




places of an inferior, and J/",, M a , &c, those of a superior planet, then will 
these latter bodies exhibit the appearances represented in the figure. 
■--•■---' § 369. Transits, Occultations, and Transit Limits. — A body which in- 
/ terposes itself between the earth and some other body, so as to conceal any 
portion of the latter from view, is said to make a transit; the masked 
body is said to be occulted, and the phenomenon is called a transit or an 
occupation, according as we refer to the masking or masked body. 

§ 370. The nodal lines of all the planets lying in the plane of the eclip- 
tic, are crossed twice a year by the earth. If at the time of crossing the 
nodal line of an inferior planet, the latter be in or near inferior conjunction, 
there will be a transit, and the planet will appear as a dark circle on the 
solar disk. 

§ 3*71. To find the greatest elongation consistent with a Transit. — 
Conceive a conical surface tangent to the sun and earth. When the planet 




96 SPHERICAL ASTRONOMY. 

at inferior conjunction passes wholly or in 
part within this surface, there will be a 
transit visible from some place on the 
earth. 

Let S be the sun, E the earth, and P 
the planet just touching the conical sur- 
face, of which AB and A B' are sections, by a plane through the centres 
of the three bodies. Make 

SEA = 8 — sun's apparent serai-diameter; 

TEP = d = planet's apparent semi-diameter ; 

E A B = * = sun's horizontal parallax ; 

E TB = *g' = planet's horizontal parallax ; 

SEP = s = planet's elongation at the beginning of the transit; 

dien will 

s = 8 + d + AET, 



but 






A ET =*'-*, 


whence 






s — 8 -\- d -f- <x' — if 



(132) 

that is, wheD the elongation of an inferior planet is less than the sum of 
the apparent semi-diameter of the sun and planet, augmented by the differ- 
ence of their horizontal parallaxes, there will be a transit or an occultation 
of the planet, according as its horizontal parallax is greater or less than 
that of the sun. 

§ 372. Let EE' be M* 77. 

an arc of the ecliptic, 
O r an arc of the plan- 
et's orbit, and JV the 
node. Parallel to 0', 
and at a distance from 
it equal to s, draw on 
either side a line cutting the ecliptic in £. 

Now, if at the time of inferior conjunction the difference between the 
geocentric longitudes of the sun and node be less than S JY, there must be 
a transit ; if greater, there can be none. The distance aS'-ZV" is called a 
transit limit. 

To find its value, make 

(SiVP = i = inclination of the planet's orbit; 
JUS = I = transit limit ; 




PLANETS. 97 

then, in the right-angled triangle S P N, 

sin ^ s _^4 (L33) 

sin % v ' 

The value of s is variable, being a function of the radii vectors of the 
earth and planet at inferior conjunction. The inclination i is also slightly- 
variable. The greatest value of i and least value of s make I a minimum 
limit ; the least value of i and greatest value of s make / a maximum 
limit 

§ 373. The earth returns sensibly to the same place of the heavens at 
intervals of a sidereal year. Any entire number of sidereal years which 
will contain the synodic revolution of a planet a whole number of times, 
will bring the earth and planet to the same places they simultaneously oc- 
cupied before, and if a transit occur at one node, it will occur at the same 
node again at the expiration of this interval, provided the node be not 
carried by its proper motion beyond the transit limit. 

§ 374. The bodies having returned to the places they previously occu- 
pied, will each have performed a whole number of entire revolutions, and 
making 

n = the number of the earth's revolutions ; 
n r == •■ " " planet's " 

P = the length of the earth's sidereal year ; 
P'= " " planet's " " 



we shall have 
whence 



nP = n'P' 

n__P^_ 
n~'~~P 



(134) 



If P and P r be whole numbers, and the second member be reduced to its 
simplest terms, the numerator will be the interval in sidereal years between 
the consecutive transits at the same node, and this interval will be constant. 

But if P and P r be not whole numbers, then will the numerators of the 
approximating fractions of the continued fraction, which give the values of 
the second member within the transit limits, be the variable intervals, in 
sidereal years, between the transits at the same node. 

§ 375. Masses and Densities of the Planets. — The masses of such of the 
planets as have satellites may easily be found by the process of § 313, as 
soon as the periodic time of the planet and that of its satellite are deter- 
mined by observation. But for such as have no satellites, recourse is had 
to a differeut process, which can be here indicated only in outline, A 

7 



98 SPHERICAL ASTRONOMY. 

planet undisturbed by the action of the others, would describe accurately 
its elliptical orbit about the common centre of inertia due to its own mass 
and that of the sun ; and from the elliptical elements already described, 
its future places are, as we shall see, predicted with the greatest precision. 
Tha difference between these places and those actually observed, give the 
effects of the disturbing action of the other planets. To compute these 
effects, what are called perturbating functions are constructed upon the 
principles of mechanics. The masses of the perturbating or disturbing 
bodies enter these functions ; and from the observed amount of perturb- 
ations the value of the masses are computed. An. Mec, § 203, 

§ 376. The masses and volumes being known, the densities result from 
the process of § 314. 

§ 3*77. Rotary motions. — All the planets whose surfaces exhibit through 
the telescope distinct marks, are found to have a rotary motion in the same 
direction as those of the sun and earth, viz., from west to east. 

§ 378. Planetary Atmosphere. — The existence of an atmosphere about 
a planet is indicated by the apparent displacement it occasions in the geo- 
centric place of a star by refracting its light, when, by the motion of the 
earth and planet, the latter comes near the line of the star and observer. 

The atmosphere about a planet is in fact a vast spherical lens, of which 
the central part is deprived of its transparency by the opaque materials of 
the planet, but of which the outer portion is free from obstruction and acts 
upon the light which passes through it with an energy due to its refractive 
power and density. 

The height of the atmosphere is inferred from the greater or less angular 
distance between the star and planet when the displacement begins ; and 
the density, which must be regulated by the same laws that govern the 
equilibrium of heavy elastic fluids upon the earth, from the amount of dis- 
placement. 

§ 379. In detailing the physical peculiarities of the planets, their mean 
distances and times of sidereal revolutions, although contained in the sy- 
noptical table of elements, will be repeated ; and in all cases in which di- 
mensions or measures are given, they must be understood as expressed in 
the corresponding elements of the earth as unity. Thus, if it be the mean 
distance, density, volume, solar heat and light, sidereal day, &c, those of 
the earth are the respective units, 



MERCURY. 99 



MERCURY. 



£ 3£0. Proceeding outwards from the sun, Mercury is the first known 
planet. His mean distance is 0.38*70985 ; sidereal year, 0.2408; true di- 
ameter, 0.398; volume, O.0G3 ; mass, 0.175; density, 2.7S r approaching 
that of gold; intensity of its attraction for a unit of mass on its suiface, 
c died surface gravitation, 1,15 ; solar heat and light, 6.68 ; time of rota- 
tion upon its axis, called sidereal day, 1 .20833. 

The eccentricity of his orbit being huge, his greatest elongation varies 
from 16° 12' to 28° 48'. The latter being his greatest apparent distance 
from the sun, he is generally lost to us in the light of that body, and it is 
difficult, therefore, to observe him. Ilis arc of retrogradation varies from 
9° 22 / to 15° 44'. 

§ 381. When to the west of the sun he lises before, and when to the 
east he sets after that luminary. In the foimer position he is called a 
morning, and in the latter an evening star. 

§ 382. The sun appears nearly seven times as large to the inhabitants 
of Mercury as to us ; and on the supposition that the intensity of solar 
light and heat varies inversely as the square of the distance, the solar il- 
lumination and temperature on Mercury would be 0.68 times that on the 
earth, as above. Heat and cold are, however, but relative terms, depending 
upon physical conditions as well as distance, and the Mercuiian surface 
may be as cold as the earth's: the frosty summits of the Himalayas are 
nearer to the sun than the scorching plains of Hindostan. 

§ 383. The changes of seasons on Mercury, depending, as they do, 
upon the inclination of his axis to that of his orbit, which has not been 
well determined, ara not accurately known. If, as there are reasons to 
believe, this inclination have any considerable value, the mutations of Mer- 
cury's seasons must be very great; his tropical year being only about one- 
fourth that of the earth, his seasons, if they follow the same propoition, 
can only be of some two or three weeks' duration. 

§ 384. Mercury's nodes are, and will for r.ges continue, in that part of 

the ecliptic which the earth passes in May and November, and his transits 

over the sun must occur in those montlis. His periodic time = 87 d .97, 

and that of the earth = 365 d .256, in Eq. (134), give the approximating 

fractions, 

7 13 33 

29 '' 54 ; 137 *' 

So that the intervals between the transits which may be expected at the 
same node are seven, thirteen, &c, years. The great inclination of Mercu- 



100 SPHERICAL ASTRONOMY. 

fy's orbil makes his transit limits, Eq. (133), small, and the above interval 
will not therefore always be those which separate the actual recurrence 9 
the transits. The last transit occurred at the ascending node in 1848 
the nest will occur in 1861. 



VENUS. 

§ 385. Venus follows Mercury in the order from the centre. Her mean 
distance is 0.723331V; sidereal year, 0.6152; true diameter, 0.975; vol- 
ume, 0.927; mass, 0.885; density, 0.95; surface gravitation, 0.93; solar 
heat and light, 1.91 ; sidereal day, 0.97315. 

§ 386. Venus is the brightest of the planets, her light being of a bril- 
liant white, and at times so intense as to cause a shadow. The elongations 
of her stations vary but little from 29°. Her phases are finely exhibited 
through the telescope. The southern horn of her crescent varies its shape, 
being alternately sharp and blunt, and the changes are attributed to the 
periodical interposition of high mountains by an axial rotation of Venus 
so as to intercept the solar light she at other times reflects to the earth 
from her southern surface. From these changes her sidereal day has been 
determined. 

§ 387. Her axis is inclined to that of her ecliptic under an angle of 
75°, thus placing her tropics at the distance of 15° from her poles, and 
her polar circles at the same distance from her equator. Her seasons suc- 
ceed each other, therefore, very rapidly, there being two summers and two 
winters in each of her annual revolutions. Her atmosphere resembles in 
extent and density that of the earth. 

§ 388. Her synodical revolution is 583.92 days. Venus is, therefore, 
about 292 days continuously to the east, and as long to the west of the sun. 
In the former position she sets after the sun, and is called an evening star; 
in the latter, she rises before the sun, and is called a morning star. Her 
greatest elongation is about 45°, and she is brightest when on her way 
from the east to the west of the sun, and at an elongation on either side of 
about 40°. 

§ 389. The line of Venus's nodes lies in that part of the ecliptic through 
which the earth passes in June and December, and her transits occur in 
those months. The periodic time of Venus = 224 d .7, and that of the 
earth = 365 d .256, which, in Eq. (134), give the approximating fractions, 

8 235 713 

— , , , &c. 

13' 382 1 1159' 



VENUS, 



101 






Fig. 78. 



*^^% 




and the transits at the same node may be expected at inteivals of eight, 
two hundred and thirty-five, &c, years. Two transits, separated by an 
interval of eight years, will occur at one node, and then at the opposite 
node after an interval of one hundred and five, or one hundred and twenty - 
two years, between the last of the first pair and first of the second pair. 

As astronomical phenomena the transits of Venus are of the highest ira 
portance. They afford the best means of ascertaining the sun's horizontal 
parallax, and therefore the earth's distance from the sun, and the dimen- 
sions of the solar system, expressed in terms of some known terrestrial 
measure. 

§ 390. The principle on which the sun's horizontal parallax is found 
from a transit of Venus may be thus illustrated. 

Conceive two observers situ- 
ated at the opposite extremities 
A and B of the earth's diameter, 
which is perpendicular to the 
plane of the planet's orbit. To 
the observer A, the planet would 

appear to transit the sun's disk along the chord m n, and to the observer 
B, along the chord p q, being the intersections of the solar disk by two 
planes through the portion D C of Venus's orbit, described during the 
transit, and each observer. A third plane through the observers and Ve- 
nus's centre would cut from the other two the lines A a and Bb, and from 
the sun's disk the perpendicular distance a b between the chords. Now, 
because the angle A V B is equal to the angle a Vb, A B will be to a b as 
Venus's distance from the earth is to her distance from the sun ; that is 
§ 385, as 1 — 0.723 : 0.723, or as 1 to 2,61 nearly ; and the radius of 
the earth, half of A B is to a 6, as 1 to 5.22 nearly. The apparent mag 
oitudes of two objects, viewed at the same distance, being directly pro- 
portional to the true magnitudes, the ra- 
dius of the earth viewed at the distance 
of the sun, in other words, the sun's hori- 
zontal parallax, is equal to the angular 
distance between the chords divided by 
5.22. 

§ 391. The relative geocentric motion 
of the sun and planet into the oh.scnrd 
durations of the transit at the two stations 
will give the chords m n and p g. The 
chords being known, as also the apparent 



Fig. 79. 




102 SPHERICAL ASTRONOMY. 

semi-diameters £ q and S n, the distances S a and & b become known, 
and therefore their difference a b. 

§ 392. The general result of all the observations made on the transit 
of 17G9 gives 8".5776 for the sun's horizontal parallax. The next two 
transits of Venus will occur on Dec. 8th, 1874, and Dec. 6th, 1882. 

MARS. 

§ 393. Mars is the first of the superior planets, His mean distance is 
1.5237; sidereal year, 1.8807; true diameter, 0.517; volume 0.1380; 
density, 0.95 ; equatorial gmvitation, 0.493 ; solar heat and light, 0.43 ; 
sidereal day, 1.02691; oblateness, about 19; and the inclination of his 
axis to that of his ecliptic 30° 18' 10".8. 

§ 394. He has a dense atmosphere of moderate height. His surface 
(Plate II., Fig. 2) exhibits through the telescope outlines of what are 
deemed to be continents and seas, the former being distinguished by a 
ruddy color, which is characteiistic of this planet, and indicates an ochry 
tinge in the soil, contrasted with which the seas appear of a greenish hue. 

These markings are not always equally distinct ; and the variation is 
attributed to the formation of clouds and mists in the planet's atmosphere. 
Brilliant white spots sometimes appear at that pole which is just emeiging 
from the long night of its polar winter, and are attributed to extensive 
snow- fields that push their borders to an average distance of some six de- 
grees from either pole. 

PLANETOIDS. 

§ 395. Next to Mars come the class of small planets, which, on account 
of their comparatively diminutive size, are called planetoids. Little is 
known of them beyond their orbit elements, but they are interesting on 
account of their history and the speculations connected with their discov- 
ery, which began with the present century. 

§ 396. If the mean distance of Mercury be taken from the mean dis- 
tances of the other planets, the remainders will form a series of numbers 
doubling upon each other in proceeding outward from the sun. To this 
law there was a remarkable exception in the distance between the orbits 
of Mercury and Jupiter as compared with that between Mercury and Mars, 
the former being so large as to require the interpolation ol another body 
between Mars and Jupiter. 

§ 397. Although th - law is strictly empirical and wholly inexplicable 



Plate HI. 




/■" 



TOTKOWT TAGE 102. 



^*Y^ 



/ 



PLANETOIDS. 103 

a priori upon any known physical hypothesis, yet the coincidence was so 
remarkable as to induce the prediction that by proper search a planet 
would be found in the interpolated place. 

§ 398. This body was only to be recognized by its proper motion. TV 
detect this, an examination of the telescopic stars of the Zodiac was com- 
menced, their places were carefully mapped, and on the first day of the 
present century, the prediction was verified by the addition of Ceres to the 
system. Her mean distance is 2.76692, and the hiatus was filled. 

§ 399. But the discovery of Ceres was soon followed by that of Pallas, 
at the mean distance of 2.7728 — nearly the same as that of Ceres — and 
\ the law was again broken. 

§ 400. The points in which the paths of the new planets are intersected, 
du either side of the sun, by the line common to the planes of both orbits, 
are not very far apart, and it was suggested that Ceres and Pallas were 
but fragments of a larger planet that once revolved at an average distance, 
and which had been broken to pieces by some disruptive force. But where 
were the other fragments ? 

§ 401. A number of bodies projected in different directions from a com- 
mon point, would each describe about the sun an hyperbola, a parabola, or 
an ellipse, depending upon the relations between the velocity of projection 
and the intensity of the sun's attraction upon the unit of mass, and in the 
case of elliptical orbits, the bodies would, abating the effects of the pertur- 
bating action of the other planets, return at fixed intervals to the place of 
departure. 

§ 402. The opposite points of the heavens, in which the orbits of Ceres 
' and Pallas approached most nearly each other, were therefore regarded as 
the common haunts of the suspected fragments, and the places especially 
to be watched, to detect their existence. A constant scrutiny of these 
points, and diligent revision of the maps of the zodiac, have resulted in the 
discovery, to the present time, of 91 of these little bodies. 

§ 403. The mean distances of the planetoids vary about from 2.2 to 3.6, 
and periodic times about from 3.3 to 6.9. Their small size makes it diffi- 
cult to determine their true dimensions, the diametor of the same individ- 
ual, as given by the best authorities, varying from 0.02 to 0.20. They 
exhibit considerable variety of color; some have shown signs of possessing 
atmospheres, and those who regard them as debris of a single body, rind 
evidence of an angular or fragments] figure in sudden changes of illumina- 
tion, which have been observed, and which are attributed to the shifting 
of their bounding planes by a diurnal or axial rotation. 



104 SPHERICAL ASTRONOMY. 



JUPITER. 



5 404. Jupiter js the largest, and except Venus, which he sometimes 
surpasses in this respect, the brightest of the planets. His mean distance 
is 5.202 ; sidereal year, 11.86; diameter, 11,2 ; volume, 1280.9 ; mass, 
331.57 ; density, 0.24— but little greater than that of water; equatorial 
gravitation, 2.716; solar heat and light, 0.037; sidereal day, 0.41376; 
oblateness, 20 ; inclination of axis to that of his ecliptic, 3° 5' 30". 

§ 405. The disk of Jupiter is always crossed, in a direction parallel to 
bis equator, by dark bands or belts, presenting the appearance indicated in 
Plate III., fig. 3, which was taken by Sir John Herschel. These belts are 
not always the same, but vary in breadth and situation, though never in 
direction. They have sometimes been seen broken up and distributed over 
the whole face of the planet. From their parallelism to Jupiter's equator, 
their occasional variation and the appearance of spots upon them, it is in- 
ferred that they exist in the planet's atmosphere, and are composed of 
extensive tracts of clouds, formed by his trade-winds, which, from the great 
size of Jupiter, and the rapidity of his axial rotation, are much more de- 
cided and regular than those of the earth. 

§ 406. The great oblateness of this planet is due to the shortness of his 
sidereal day, and its amount agrees with that assigned by theory to give 
him a figure of fluid equilibrium. 

§ 407. From the small inclination of his axis to that of his ecliptic, there 
can be but little variation in the length of his days and nights, each of 
which is less than five of our hours ; and changes of seasons must be 
almost, if not quite unknown to his inhabitants. 

§ 408. Jupiter is attended in his circuit about the sun hy four satellites 
or moons, which revolve about him from west to east, and present a min- 
iature system analogous to that of which Jupiter himself is but a single in- 
dividual, thus affording a most striking illustration of the effects of gravi- 
tation and of distance in grouping, as well as shaping the courses of the 
heavenly bodies. These satellites will be noticed under the head of Sec- 
ondary Planets. 

SATURN. 

§ 409. Saturn is the next in order of size as he is of distance to Ju- 
piter. His mean distance is 9.538850 ; sidereal year, 29.46 ; true diam- 
eter, 9.982 ; volume, 995.00; mass, 101.068 ; density, 0.102— little more 
than half that of water; equatorial gravitation, 1.014; solar heat and 



Plate W. 




TO FRONT PAG-E 104 



Plate V. 




TO FRONT PAGE 105 . 



SATURN. 105 

light, 0.011 ; sidereal day, 0.43701 ; oblateuess, 25; inclination of axis 
to that of orbit, 26° 49', and to that of our ecliptic, 28° 11'. 

§ 410. Saturn is the most curious and interesting body of the system, 
being attended by eight satellites or moons, and surrounded (Plate IV., Fig. 
4), according to some authorities by two, and others by four, broad flat and 
extremely thin rings, concentric with each other and with the planet. 

§ 411. The dimensions of the rings and planet, and the intervals as 
given by the advocates of but two rings, are, 

n miles. 

Exterior diameter of exterior ring .... 40.095 = 176,418 
Interior " " " . . . . 35.289 == 155,272 

Exterior diameter of interior ring .... 34.475 = 151,690 
Interior " " " . . . . 26.668=117,339 

Equatorial diameter of planet 17.991=: 79,160 

Interval between the planet and interior ring 4.339= 19,090 

Interval between the rings 0.408 = 1,791 

Thickness of ring not exceeding 230 

§ 412. The evidence of recent observations with very powerful instru 
ments seems, however, in favor of a division of the outer ring, as just given, 
at a distance less than half its width from the exterior edge, and of the 
existence of a dusky ring still nearer the body of the planet, and composed 
of materials partially transparent, and possessing but feeble powers of re- 
flection, resembling in these particulars a sheet of water. And there seem 
good reasons for believing that the rings are not precisely in the same 
plane. 

The disk of the planet is crossed by parallel belts, similar to those of 
Jupiter ; these are supposed to be due to Saturn's trade-winds. From the 
parallelism of the belts to the plane of the rings, it is inferred that the 
planet's axis of rotation is perpendicular to that plane, and this is con- 
firmed by the occasional appearance of extensive dusky spots on his sur- 
face, which, when carefully watched, give the time of his rotation about 
an axis having that direction. 

§ 413. By watching the different shades of illumination on different 
portions of the rings, the latter are found to complete a revolution in 
their own plane once in 10 h 32 m 15", thus making their sidereal day 
0.43906, which exceeds that of the planet itself by 0.00205. 

§ 414. That the rings are opaque and non-luminous is shown by their 
throwing a shadow on the body of the planet on the side nearest the sun, 
and bv the other side receiving that of the planet as shown in the ti^'ure. 



106 



SPHERICAL ASTRONOMY 

Fig. 80. 




§ 415. The axes of the planet and rings preserve their directions un- 
changed during their orbital motion. The plane of the rings, which is 
inclined to that of the ecliptic under an angle of 31° 19', intersects the 
latter plane in a line which makes with the line of the equinoxes an angle 
equal to 167° 31', so that the nodes of the ring lie in longitudes 167° 31' 
and 347° 31'. 

§ 416. The orbital motion of the planet causes this intersection to oscil- 
late, as it were, parallel to itself, in the plane of the ecliptic, through a 
distance on either side of the sun equal to the radius vector of Saturn's 
orbit ; and the period of a semi-oscillation is one-half of the planet's pe- 
riod, or about 15 years. Within this period the plane of the ring must pass 
once through the sun, and from once to thrice through the earth, depend- 
ing upon the initial position or place of the latter when the trace of the 
plane on the ecliptic touches the earth's orbit at the time of nearing 
the sun. 

§ 417. Thus, let S be the sun, EE'E"E'" the earth's orbit, P P' an 
arc of Saturn's orbit projected upon the plane of the 
ecliptic, P E and P' E" the traces of the plane of 
the rings on the same, and tangent to the earth's 
orbit, and suppose the motion of the earth and of 
Saturn to take place in the direction indicated by the 
arrow-heads. Draw SB parallel to P E and P' E", 
and make 

r = S P = the mean distance of Saturn ; 
r'= SE = " " " of earth; 
a = P S P' = the angle at the sun subtended by 
PP' : 

then, since the angle P S B = S P E, we have 

• i r ' l 

sin \ a = — = — - 

J r 9.54 

whence a = 12° 2' 



Fig. 81. 



-^m^j?. 



0.1082, 




SATURN. 107 

which divided by 2' 0".6, the mean motion of Saturn, gives 350.46 days, 
wanting only 5.8 days of a complete year ; that is to say, the earth de- 
scribes nearly one entire revolution in the time during which the earth's 
orbit is traversed by the plane of the ring. 

§ 418. The rings are invisible when their plane passes between the sun 
and earth, their enlightened face being then turned from the latter body ; 
and the interval of non-appearance will be that between any two epochs 
at which the plane passes the sun and earth, and of which the effect of 
one is to throw these bodies on opposite and the other to restore them to 
the same side of this plaue. 

§ 419. If the initial place of the earth be at E ir , nearly three days in 
advance of B" , then will the plane itself pass the sun and earth at the 
same time, the earth being at B f , and these bodies could not be on oppo- 
site sides of the plane of the rings during its present visit to the earth's 
orbit. If the initial position of the earth be at E\ nearly three days in 
advance of E, it will be at E" when the plane passes the sun ; the rings 
will then disappear, and continue invisible till the earth meets and passes 
their advancing plane, which it will do somewhere in the quadrant E" B' \ 
they will then reappear, and continue visible for the next fifteen years. If 
the earth's initial place be at E'", some days in advance of B\ it will 
meet and pass the plane in the same quadrant, the rings will disappear and / V 
continue invisible till their plane is overtaken and passed again by the 
earth somewhere in the quadrant E B" ; when the plane passes the sun 
the earth will be in the quadrant B"E", and the rings will again disap- 
pear, and again become visible only when their plane is recrossed by the 
earth in the quadrant E"B' . Thus, with this initial place, the earth will 
cross the plane of the rings three times in one year, and there will be two 
disappearances. 

§ 420. When the plane of the ring passes through the sun, the edge 
of the ring alone is enlightened, and can only appear as a straight line of 
light projecting from opposite sides of the planet in the plane of his equa- 
tor, and parallel to his belts. This phase of the ring has been seen, but it 
requires the most powerful telescopes ; and from the fact of its non-ap- 
pearance in a telescope which would measure a line of light one-twentieth * f 
of a second in breadth, of which the subtense at Saturn's distance is 230 
miles, it is inferred that the thickness of the ring cannot exceed this latter 
dimension. ** 

§ 421. When the dark side of the ring is turned to the earth, the 
planet appears as a bright round disk with its belts, and crossed equato- 
rial ly by a narrow and perfectly black line. This can only happen when 



v 




10g SPHERICAL ASTRONOMY. 

the planet is less than 6° 1' from the node of his rings. Generally th6 
northern side is enlightened when the heliocentric longitude of Saturn is 
between 172° 32' and 341° 30', and the southern when between 353° 32' 
and 161° 30'. The greatest opening occurs when the heliocentric longi- 
tude of the planet is 77° 31' or 257° 31'. 

URANUS. 

§ 422. Uranus is one of the more recently discovered planets, being 
only recognized as a planet for the first time in 1*781, though it had 
often been seen before and mistaken for a fixed star. 

Of this planet nothing can be seen but a small round uniformly illumi- 
nated disk without rings, belts, or discernible spots. His mean distance is 
19.18239; sidereal year, 84.01; true diameter, 4.36; volume, 82.91; 
mass, 14.25 ; density, 0.17 ; equatorial gravitation, 0.75 ; solar heat and 
light, 0.003. He is attended by six satellites, which will be noticed 
presently. 



VV § 423. 
^xVH^ird in si 




NEPTUNE. 

Neptune is the last known planet in the order of distance, and 
size. Its discovery dates only from 1846, though its existence had 
been suspected from certain irregularities in the motion of Uranus, which 
could only be attributed to the disturbing action of some body exterior to 
itself. 

The departures of Uranus from places assigned by the combined action 
of the known bodies of the system, and certain assumed conditions in re- 
gard to position and shape of orbit, direction of motion, and mean distance, 
rendered highly probable by analogy, were the data from which, by the 
methods of physical astronomy, was wrought out in the closet in Paris, 
the place of a new planet whose disturbing action would account for 
the unexplained waywardness of Uranus. The result was sent to an 
observer in Berlin, and in the evening of the very day of its receipt in 
the latter city, Neptune was added to the known system by actual 
observation. It was found within 52' of the place assigned, and its 
discovery, in all its circumstances, must ever be regarded as one of the 
greatest triumphs of modern science. 

§ 424. Neptune's mean distance is 30.0367 ; periodic time, 164.6181 ; 
real diameter, 4.5 ; volume, 91.125; mass, 18.219 ; density, 0.208 ; equa- 
torial gravitation, 0.9035 ; solar heat and light, 0.0011. 

The apparent size of the sun as seen from the earth, bears to that as seen 



SECONDARY BODIES. 109 

from Nep'une, about the relation of an ordinary orange to a common duck- 
shot. 

§ 425. Neptune has at least one satellite, and certain appearances have 
indicated a second, and also a ring, but of these there are yet doubts. 

General Remark. 

§ 426. In the foregoing enumeration of the physical peculiarities of the 
planets, one is impressed by the great differences in their respective sup- 
plies of heat and light from the sun ; in the relations which the inertia oi 
matter bears to its weight at their surfaces ; and in the nature of the ma- 
terials of which they are composed, as inferred from variety of mean 
density. The intensity of solar radiation is nearly seven times greater on 
Mercury than on the earth, and on Neptune 900 times less, giving a range 
of which the extremes have the ratio of 6300 to 1. The efficacy of weight 
in counteracting muscular effort and repressing animal activity on the 
earth, is less than half that on Jupiter, more than twice that on Mars, and 
probably more than twenty times that on the planetoids, making a range 
of which the limits are as 40 to 1. Lastly, the density of Saturn does not 
exceed that of common cork. Now, under the various combinations of 
elements so important as these, what an immense diversity must exist in 
the conditions of animal life, if the planets, like our earth, which teems 
with living beings in every corner, be inhabited ! A globe whose surface 
is seven times hotter than ours or 900 times colder, on which a man might 
by a single muscular effort spring fifty feet high, or with difficulty lift his 
foot from the ground ; where his veins would burst from deficiency or col- 
lapse from excess of atmospheric pressure, affords to our ideas an inhospi- 
table abode for animated beings. But we should remember that heat and 
cold, light and darkness, strength and weakness, weight and levity, are but 
relative terms ; and to the very conditions which convey to our minds only 
images of gloom and horror, may be adjusted an animal and intellectual 
existence which make them the most perfect displays of wisdom and be- 
neficence. 

SECONDARY BODIES. 

§ 427. The secondary bodies are those which revolve about the planets, 
and accompany them around the sun. Of these, twenty are known at the 
present time. One belongs to the earth, four to Jupiter, eight to Saturn, 
six to Uranus, and one to Neptune. They are commonly called satellites, 
and sometimes moons, but this latter appellation is more particularly ap- 
plied to the earth's secondary. 



THE MOOK 

§ 428. The moon revolves in an elliptical orbit, of which one of the 
f oei is at the earth's centre. Its motion is from west to east, and its an- 
gular velocity about the earth is much greater than that of the earth 
around the sun. «The moon appears, therefore, to move among the fixed 
stars in the same direction as the sun, but more rapidly ; and from the 
axial motion of the earth she has, like other heavenly bodies, an apparent 
diurnal motion, by which she rises in the east, passes the meridian, and 
sets in the west. 

§ 429. The oblateness of the earth would be quite appreciable to an ob- 
server at the distance of the moon. Her equatorial horizontal parallax is 
therefore found from Eq. (24); her distance from Eq. (28); her true diam- 
eter from Eq. (29) ; and her mass from her effects in producing precession 
and nutation. 



Lunar Orbit. 



430. The elements of the moon's orbit may be found from four ob- 
served right ascensions and declinations, corrected for refraction, parallax, 
and semi-diameter. 

Fig. 82. 



c-JiL 



Let D C be an arc in which 
the plane of the orbit cuts the 
celestial sphere ; V B an arc of 
the ecliptic, and V A of the 
equinoctial ; V the vernal equi- 
nox, N the ascending node, P 
the perigee, and M h M 2 , M,. 
M A the geocentric places of the 
moon. 

First convert the geocentric 
right ascensions and declina- 
tions into geocentric longitudes 
nnd latitudes, and make 

v ~ VN = longitude of node ; 
i — CjVJB = inclination of orbit 
l x = V O x = longitude of M x ; 




l 2 = V 2 
X, ±r M x O x 
X2 = M t 0, 



__ u 



M 2 
latitude of M x ; 




^^^^-E?-r^T^r 





^y ^— THE MOON. 

thenln the right-angled triangles M x N O x and M 2 N 2 , we have 

sin (/, — v) = cot i . tan \ ) /fH35) 

sin (4 — v) = cot i . tan X 2 j ' /^' ,// 

and by division C^/^y^ S — ; / 

sin (l L - v) = tanX, K^^Z^^V^~^t 

sin (4 — v) tan X 2 ' 

Adding unity to both members, reducing to common denominator, then 
subtracting each member from unity, reducing as before, and finally divi- 
ding one result by the other, we find 

sin (/ 2 — v) -f- sin (l x — v) tan X 2 + tan X, } 
sin (/ 2 — v) — sin (l x — v) — tan X 2 — tan X t ' \ 

replacing the members by their equals, we have 



f 



[h-hh 1 1/7 7-, sin (X, 4- X.) .„ „ 

X__i_ v ] = t an i (4-A). i ^-^ • (13C, 

Also, from first of Eqs. (135), we have 

sin (I, — v) , 

coti= V1 . — I (137) 

tan X t v ' 

whence v and i are known. 

The longitude of the ascending node, increased by the angular dis- 
tance of a body from the same node, is called the Orbit Longitude. 

Make 

v x — VEN + NEM X = orbit longitude of M x ; 

p= VEJST+JVEP = " " perigee; 

9 = PEMi = v x —p = true anomaly of M x ; 
e = eccentricity of orbit ; 

m =. mean motion of moon in orbit; 

t x = time since epoch for M x ; 

L = mean orbit longitude at epoch. 

Then resuming Eq. (48), we have 

L -f- mti = v x — 2e sin (v x — p) . . . . (138) 



in which 

v x s, + tan-' '"' ■ . ' 7 .... (139) 



., tan (/, — v) 



cos 



Four values for the ge )centric longitudes denoted by l h l a , l s , l 4l in Eq. 
(139), give four values for v, viz., v u v 9i v 3l and v 4 ; and these, and the time* 






s 
1X2 ' SPHERICAL ASTRONOMY. 

of observation t h t 2 , t 3 , and t 4i in Eq. (1-38), give four equations involving 
the four unknown quantities Z, ??i, e, and p ; whenee these become known 
precisely as in § 197, employing for the purpose Eqs. (50), (51), (53), 
and (54). 

§ 431. Denoting the ecliptic longitude VO of the perigee by^, we 
have, in the triangle N P 0, right-angled at 0, 

tan NO = tan (p — v) . cos i, 
and 

pi = v + tan"" 1 [tan (p — v) . cos i\ . . . (140) 

§ 432. In the same way, denoting the mean ecliptic longitude of the 
moon at the epoch by Z„ 

Z/[ = v + tan -1 [tan (L — v) . cos i] . . . . (141) 

§ 433. The passage of the moon through one entire circuit of 360° 
around the earth, is called a sidereal revolution. The interval of time re- 
quired to perform a sidereal revolution is called a sidereal period. Denote 
the sidereal period by s, then will 

360° 



(142) 



m 

The equation of the orbit, the centre of the earth being the pole, is 

a (1 - e 8 ) 

f — 1 L • 

1 -f e cos (v — p) ' 

and the value of r being found by means of Eq. (28), that of the mean 
distance a will result, and every thing in regard to the moon's path be- 
comes known. 

§ 434. At the epoch January 1st, 1801, the elements of the lunar orbit 

were 

Mean a = 59.96435000 of the earth's equatorial radius; 

" s =: 27.321661418 mean solar days; 

" e— 0.054844200; 

« v= 13° 53' l7".7; 

" ^ = 266° 10' 07".5; 

" i= 5° 08' 47".9; 

" Z, = 118° 17' 08".3. 

'§ 435. The moon's true diameter, Eq. (29), is 0.27280, or about 2153 
miles; volume, 0.0204; mass, 0.01 13S9; density, 0.5657; and surface 
gravitation, 0.1666. 

§ 436. Comparing the lunar elements which depend upon the orbit as 



THE MOON 



113 



determined at different times,, they are all found to vaiy. The nodes have 
a retrograde and the perigee a afreet motion, the former performing a com- 
plete revolution in 18.6, and the latter in 8.854 years. The inclination 
fluctuates between 4° 51' 22" and 5° 20' 06" ; the mean distance has a 
secular variation, and it is at the present time diminishing ; the same is 
true of the sidereal revolution, and the mean motion of the moon is increas- 
ing. All these changes are due to the disturbing action of the other bod- 
ies of the system, but principally of the sun. The action of the protuberant 
ring of matter about the equator of the earth also has its effect. 



Disturbing Forces. 

§ 437. To illustrate the way in which Fi s- & 

these changes of the lunar orbit are 
brought about, let E be the earth, S the 
sun, M the moon, moving in her orbit in 
the direction MDN'N; N and N' be- 
ing the nodes, and E V the direction of 
the vernal equinox. Then, resuming 
equations (80) and (81), making p — 
EM, the radius vector of the moon, and 
employing in all other respects the nota- 
tion of § 286, v becomes the change 
which the sun's attraction causes in the 
weight of a unit of the moon's mass due 
to the earth's attraction, and <r the 
change which the sun's attraction causes 
in the force normal to the radius vector 
and in the plane passing through the sun, 
earth, and moon. This latter force being 
in general oblique to the plane of the lu- 
nar orbit, urges the moon out of that 
plane, and causes her to describe a curve 
of double curvature, while the former has 
no such action. 

Resolve <r into two components, one perpendicular to the radius vector 
and in the plane of the orbit, the other normal to this latter plane. For 
this purpose conceive a sphere of which the centre is at that of the earth, 
and radius, the radius vector p = EM, of the moon. Its surface will be 
cut by the plane of the ecliptic in AN 4 BN 4J , by that of the lunar orbit 

8 







114 



J 

SPHERICAL ASTRONOMY 



in N t MN U , and by that of the sun, 
earth, and moon in AM B. Make 

= V E S = sun's longitude ; 
ft = VEjy,= long, of moon's node; 

i = MN t A = inclination of lunar or- 
bit; 

X = N t MA = inclination of orbit to 
plane of sun, earth, 
and moon ; 

ic = <r . cos X = component of r in 
plane of lunar orbit ; 

o = <r . sin X = component normal t6 
plane of orbit. 

In the triangle N i M A, the arc N { A 
= (0 — ft) and AM = <p, and we have 



/-* 



Fig. 83 bis. 



sin X = 



sin i .sin (O — ft) 
sin <p 

, sin 2 i. sin 9 (O — ft) 
cos 2 X = 1 — ^ ; 




IP — -B 



and bringing forward Eq. (80), and replacing r by its value in Eq. (81), 
there will result 

== —^^.(2 cos 2 <p — sin 2 <p) (143) 

O /yyi §4* f\ . , ■ 

= ~l3 • cos <p . y sin 2 9 — sin 2 i sin 2 (0 — ft) (144) 



2 ra &p 



. cos <p . sin i . sin (O — ft) 



(145) 



v is called the radial, t the transversal, and o the orthogonal disturbing 
force. These forces- acting at right angles are independent of each other, 
and affect the movements of the moon in modes perfectly distinct, which 
will become manifest by discussing the above equations. 

§ 438. The radial force being directed towards or from the earth's centre, 
changes the simple law of gravitation, and alters the elliptic form of the 
orbit, sometimes diminishing and sometimes increasing its eccentricity, 
and shifting the place of its greatest curvature, that is, the position of the 
apsides. 

§ 439. The transversal force is exerted to accelerate or retard the 



THE MOON. 

moon's motion in her orbit, and to give rise to fluctuations to and fro 
about that due to the action of the earth alone, and thus to alter the ellip- 
tic path. 

§ 440. The orthogonal force deflects the moon from the plane in 
which she would move under the undisturbed action of the earth, and 
causes her to describe a curve of double curvature. A plane through two 
consecutive elements of such a path must in general be oblique to that 
through two other consecutive elements, and these two planes must in 
general intersect the plane of the ecliptic in different lines; that is, the 
orthogonal force is effective in producing a motion of the nodes. By dis- 
cussing the value of this force, it will be found that while it causes the 
nodes to move in different directions at different times, on the whole, it 
causes them to retrograde. 

§ 441. Nothing has thus far been said of the variations in these disturb- 
ing forces arising, all other things being equal, from the change in the 
value of </, or the earth's distance from the sun. This gives rise to a still 
further complication by introducing an annual variation in the values of 
», tf, and e. 

§ 442. The other bodies of the system produce effects similar to those 
of the sun, but much less in degree. The protuberant ring of matter 
which projects beyond the sphere described upon the earth's polar axis 
has, as already remarked, its effect also ; so that the longitude of the 
moon, as determined from the elements of a true elliptic motion, must re- 
ceive from SO to 40 corrections to obtain that of her true place. 

§ 443. These corrections are called equations ; their forms are deter- 
mined by investigations in physical astronomy, and their coefficients are 
computed from the observed departures of the actual from the elliptic 
places. 

Librations. 

§ 444. The moon revolves uniformly about an axis inclined to that of 
her orbit, under an angle of 6° 38' 58", which is slightly variable ; and 
the time of one revolution is equal to her sidereal period. 

§ 445. Were her orbital motion uniform, and her axis perpendicular to 
her orbit, this equality would cause the moon always to present the same 
face to the earth. As it is, however, the visible portion of her surface is 
slightly variable, and in the course of a sidereal period we see a little more 
than a hemisphere. The changes of oibital motion cause small portions of 
her surface near her eastern and western borders to enter and depart from 
the field of view in the course of each revolution; and the inclination of 



116 



SPHERICAL ASTRONOMY. 



her axis to that of her orbit exposes to us her north or south pole alter- 
nately within the same period. These circumstances give rise to appa- 
rent oscillatory motions in the moon itself, which are called librations ; 
those due to irregularity of orbital motion are called longitudinal, and 
those which arise from inclination of axis, latitudinal librations. 

§ 446. In addition, slight variations take place in the visible portions 
of the moon's surface from changes in the observer's point of view, by the 
earth's rotation. These are called parallactic librations. 

Lunar Periods. 



§ 447. The moon's equinoctial is inclined to the ecliptic, and its ascend- 
ing node always exactly coincides with the descending node of her orbit ; 
so that the moon's axis describes a conical surface about the axis of the 
ecliptic once in 18.6 years. 

§ 448. The passage of the moon from conjunction with the sun to 
conjunction again, or from opposition to opposition, is called a synodic 
revolution. Her passage from one longitude to the same longitude again, 
a tropical revolution ; from perigee to perigee, or from apogee to apogee, 
an anomalistic revolution ; from one node to the same node again, a nodi- 
cal revolution. The intervals of time required to perform these revolutions 
are called periods. 

§ 449. To. find the length of either of Fig. 84 

these periods, say the synodic, let S be 
the sun, E the earth, M the moon in con- 
junction, E y the place of the earth at the 
next conjunction of the moon, then at M { , 
Draw E , Jf 2 parallel to E S. At the sec- 
ond conjunction the moon will have re- 
volved through 360° about the earth, in- 
creased by the angle M^E\M V — E S E x 
= the earth's angular motion in the same 
time. Make 

m = moon's mean daily motion ; 
n = earth's " 
t = synodic period. 

Then tm — 360°-|- E S E x , 

tn = ESE t , 

and by subtraction, 

tm — tn = 360°; 




THE MOON. 117 

360° ,* , 

whence t = ....... (146) 

m — n 

§ 450. Here n denotes the real angular motion of the earth, which is 
equal to the apparent angular motion of the sun. If it be replaced by 
the apparent geocentric motion of the vernal equinox, that of the apogee, 
or that of the node, taking care to give to each its appropriate sign (plus 
when the motion is direct and negative when retrograde), the correspond- 
ing period will result. The mean daily motion of the vernal equinox is 
equal to 50".2 divided by 365 d .242+ ; that of apogee to 360°, divided by 
the number of mean solar days in 8.854 years ; and that of the node by 
the number of days in 18.6 years. 

The synodic period of moon = 29.53 -J- mean solar days. 
The anomalistic " " =27.55+ " " 

The tropical " " = 2*7.32 + ." " 

The nodical " " = 27.21 + " " 

The synodic period of the moon is called a lunar month, or lunation. 



Lunar Phases. 

§ 451. The sun's distance from the earth being 23984, and that of the 
moon only 59.96 times the earth's radius, the angle at the sun subtended 
by the semi -transverse axis of the lunar orbit is 0° 08' 36"; so that rays 
of light proceeding from the sun to the moon and earth may be regarded 
as sensibly parallel ; and the exterior angle of elongation S P E\ Fig. 74, 
and Eq. (131), may be assumed equal to the tiua elongation SEP. 
Also the variation in the moon's distance is too small to produce sensible 
change in her apparent diameter to the naked eve, and the change be- 
comes perceptible only when viewed through measuring instruments. 
The apparent diameter varies from 29' 21".91 to 33' 31".07, that at the 
mean distance being 31' 07". 

§ 452. Resuming Eq. (131), making d constant, and & equal to the 
moon's elongation, and supposing the sun to the right of the figure in the 
direction of ES produced, the earth at E, and the moon successively in 
the positions 1, 2, 3, 4, 5, 6, 7, 8, we shall find the phases represented in 
the figure on next page. 

When in conjunction at 1, A or the elongation is zero, the moon is in- 
visible, anil this phase is called new moon. When at 2, the elongation 
being 45° east, the moon is said to be in first octant, and the phasf is 



Hg SPHERICAL ASTRONOMY. 




crescent When at 3, the elongatiou being 90° east, the moon is said to 
be in first quarter, and the phase is dichotomous. When at 4, the elon- 
gation being 135° east, the moon is said to be in second octant, and the 
phase is gibbous. When in opposition at 5, the elongation is 180°, the 
phase is full, and is called full moon. When at 6, the elongation being 
135° west, the moon is said to be in third octant, and the phase is gibbous. 
When at 7, the elongation being 90° west, the moon is said to be in the 
third quarter, and the phase is again dichotomous. When at 8, the elon 
gation being 45° west, the moon is said to be in fourth octant, and the 
phase is crescent. The interval of time required for the moon to pass 
through all these phases and resume them anew, is one synodic period, or 
lunation. 

§ 453. The earth presents to the moon the same phases that the moon 
does to us ; the angle of elongation of the earth, as seen from the moon, 
being always the supplement of the elongation of the moon, as seen from 
the earth. 

§ 454. The pale light of the moon, by which its outline is defined in 
conjunction, is due to the light reflected from the earth, then full, falling 
upon the dark side of the moon 

ECLIPSES OF THE SUN AND MOON. 

§ 455. The planets and sate iites, being opaque, non-luminous bodies, 
and receiving their light from the sun, which is of vastly greater size, cast 
conical shadows, of which the suifaces produced must always be tangent 
to the sun's surface. The axes of the shadows cast by the planets, lie in 
the planes of their respective orbits. That of the earth is in the plane of 
the ecliptic ; and if at the time of syzygy the moon be near one of her 
nodes, she will either pass within the luminous portion of the conical 






ECLIPSES 



119 



space between the earth and sun, or enter the earth's shadow, according 
as her phase is new or full. 

In the first case, she will mask the whole or part of the sun from some 
portions of the earth's surface ; and in the latter, will suffer a loss of the 
light she herself receives from that body. 

§ 456. The obscuration of the sun, by the interposition of the moon 
oetween the sun and earth, is called a solar eclipse. The obscuration of 
the moon, by a loss of solar illumination while within the earth's shadow, 
is called a lunar eclipse. 

Fig. 86. 




§ 45*7. Let S be the sun, E the earth, D V x and C V x tangents to the 
sun and earth ; AV V B will be the earth's shadow. Let M be the moon 
just entering the shadow, and H H' a right section of the latter at the 
distance of the moon. Make 

<k = E C A =b sun's horizontal parallax ; 

<f = CE S = sun's apparent^semi-diameter ; 
P = E H A = moon's equatorial horizontal parallax ; 

s = H E ' M — moon's apparent semi-diameter ; 
R = E A = earth's equatorial radius ; 

then in the triangle E V x C, 

angle V x = <f — if ; 

in the triangle E H V h 

angle S/z=± 18P° — P\ 
and same triangle, 

EV X : EH : : sin (180°- P) : sin jV-*); 

whence 

EV^EH. ^^-r^ EH.-^. . (147) 

sin (tf — ir) d — tr ' 

The least value for P is 52' 50" ; the greatest value for <f — * is 16' 10" ; 
whence the length of the earth's shadow is always greater than three times 
the distance of the moon. 



120 



SPHERICAL ASTRONOMY. 

Fig. 86 bis. 







§ 458. Again, in same triangle, 

HE V x = EH A -EV X H; 
and denoting the angle HE V { by E, we have 

E = p + «— a (148) 

The least value for P + it — tf is 36'' 41"; whence the apparent semi- 
diameter of the section of the earth's shadow at the moon's distance is al- 
ways greater than twice that of the moon ; and this must be increased by 
about one-fiftieth of its value for the atmospheric absorption of the solar 
light which passes near the earth's surface. The moon may, therefore, 
enter completely within the earth's shadow. 

§ 459. Denoting the angular distance V X E M between the axis of the 
earth's shadow and moon's centre, at the beginning or ending of the lunar 
eclipse, by 4 n we have 

4, = E+s = P + m—<J + s . . . . (149) 

§ 460. The conical space on the opposite side of the earth from the 
sun, and of which the bounding surface is tangent to these bodies, and 
vertex between them, is called the earth's penumbra. Thus, L B AU is 
the earth's penumbra. Its apparent semi-diameter L E Vy, at the distance 
of the moon, denoted by E„ is obtained from Eq. (148) by simply 
changing the sign of tf, these semi-diameters falling, in this case, on 
opposite sides of the axis SE; and we have 

E, = P + * + <f (150) 

The moon experiences a loss of light from the instant she touches the pe- 
numbra, and this loss continues to increase till she enters the umbra or 
shadow. 

§ 461. Now, let S be the sun, M the moon at new, E the centre of 
the earth, BA an arc of the earth's surface — enlarged to avoid confusing 
the figure. The space N V x N' is the moon's shadow, and B N' N A her 
penumbra. To all places within the section of the former by the earth's 
surface, and of which a 6 is the diameter, the sun will be totally, and to 



ECLIPSES. ~otT7cl/ 121 

i 
/ 



Fig. 87. 




CO 



all places within the annular space, of which B a and b A are sections, 
partially obscured, and present in the latter case a crescent phase. Make 

r = ME = moon's distance ; 
r, = S E = sun's distance ; 
d = MN= moon's true semi-diameter ; 
d / = S C = sun's true semi-diameter ; 
x = E V x = distance of conical vertex from earth's centre. 

Then, in the triangles V x S C and V x MJV, right-angled at C and JV", 

d,r — dr, 7* . 



whence 



up 



X = 



d, 



cCs * 



and substituting the values of r, r n d, and d„ as given by equations (28) 
and (29), 

^ tf — s 
x = R.u.-- — (151) 

§ 462. Taking the values for P, at, tf, and s, which give this the greatest 
positive and negative values, it is found that the vertex V ] sometimes falls 
short of the earth's centre about 7.6, and at others extends beyond that 
point about 3.5 times the earth's radius. 

§ 463. Again, R — x gives the distance of the vertex V u from the sec- 
tion of which a b is the diameter. Denoting this latter by y, we have, in 
the triangles a V b and N V\ N', 

r — x : R — x :: 2c? 



*,< ? f/ *; 



y\ 



y = 



2d .(R-x) 



(152) 



and substituting the values of d, x, and r, from equations (29), (1^1), and 
(28), we find 

P d — <* s — w (rf — s) 



y = 222.~ 



,(*-*> 



053) 



122 SPHERICAL ASTRONOMY. 

V 

§ 464. As long as R — x is positive, y, Eq. (152), will be positive, and 
(he shadow will reach and cut the earth. To all places within the boun- 
dary of this intersection, the sun will be wholly invisible,. and the eclipse 
is said to be total. The greatest value for y positive is about 170 miles. 

§ 465. When R — x is zero, the vertex of the shadow just comes tc 
the earth, and the sun can be totally eclipsed only to one place at a time. 

§ 466. When R — x is negative, the vertex falls short of the earth, and 
the sir. face of the latter intersects the opposite nappe of the conical shadow. 

Fig. 88. 




the value of ij is, Eq. (152), negative; it measures the distance by which 
the opposite edges of the inner boundary of the penumbra overlap one an- 
otheij and the space of which y negative is the diameter may be called the 
vmbral penumbra, and is distinguished from the rest of the penumbra in 
embracing those points from which the sun appears as an unbroken ring 
around the black disk of the moon, while to all other points of the penum- 
bra he will appear as crescent. In the first case the eclipse is said to be 
annular; in the second, crescent. The greatest possible diameter of the 
umbral penumbra is about 240 miles. 

§ 467. To find AB, the diameter of the external boundary of the pe- 
numbra on the earth, it is only necessary, to change the sign of s in Eq. 
(153), because in this case d and s fall on opposite sides of the axis of the 
moon's shadow. Making this change in Eq. (153), we have 

, l= *i:*L%£^ .... m) 

The greatest value for which is about 4835 miles. 

§ 468. The solar eclipse begins at the instant of first, and ends at the 
instant of last contact of the moon with the cone tangent to the sun and 
earth ; and the places of first and last appearance on the earth are those at 
which the corresponding rectilinear elements of this cone are tangent to its 
surface. The solar eclipse being only visible to those places situated with- 
in the path of the pmunibra, is, when considered with reference to thfe 



ECLIPSES. 153 

whole earth, called a general eclipse of the sun, in iontradistinction to its 
local character, of which more will be said presently. 

§ 469. Let M 2 (Fig. 86) be the place of the moon at the beginning of a 
general eclipse of the snn ; denote her angular distance M 2 E S from the sun 
by A ; then, since EKB — P, E V l D = 4 — «-, and M 2 EK= s, will 

4 = P + s + <J — «. . " (155) 

§ 470. In a lunar eclipse, if the moon's centre should cross the axis of 
the earth's shadow the eclipse is said to be central. When, during a solar 
eclipse, the centre of the sun, that of the moon, and the eye of the specta- 
tor are on the same right line, the eclipse is said to be central. If at time 
of syzygy the moon become tangent to the cone of the earth's shadow 
without entering, the phenomenon is called appulse. 

§ 47 1. The atmospheric lens which envelops the earth causes the solar 
light passing through it to converge to a focus between the moon and 
earth, and this light diverging anew after concentration, and falling upon 
the lunar disk while in the earth's shadow, gives to it a dark, coppery-red 
illumination, and prevents total obscuration of the moon during a lunar 

eclipse. 

* 

Relative Geocentric Orbit of the Moon. 

§ 472. The path which a body in motion appears to describe in refer- 
ence to another also in motion, is called a relative orbit ; and the distance 
of the one body from the other at any time will be the same whether we 
regard both as moving with their actual velocities, or one at rest and the 
other- moving with a velocity of which the components in any three rec- 
tangular directions are equal to the differences of the components of the 
actual velocities in the same directions. 

§ 473. Let NO be an arc of the Fi s- 89. 

ecliptic, NL an arc of the lunar orbit \Z__ 
projected upon the celestial sphere, N ^^ 

one of the nodes, S the point in which 

the axis of the earth's shadow pierces the 

celestial sphere when the moon is in her 

node, the place of this point when the moon is either in conjunction or 
opposition at L. From the node to opposition or conjunction the moon 
will have described the arc NL and the axis of the earth's shadow, whose 
motion is always equal to the apparent motion of the sun, the arc S 0. 
L is an arc of a circle of latitude ; and assuming NL to represent the 



124 



SPHERICAL ASTRONOMY. 




moon's actual velocity, NO and OL Fig. 89 bis. 

will be its components in longitude and 

latitude respectively. S is the velocity 

of the earth's shadow, which is wholly in 

longitude. If, therefore, we construct 

upon NS = NO-,SO, and SM = 

L 0, the rectangle SB, the diagonal iVifwill be an arc of the relative 

orbit of the moon referred to the axis of the earth's shadow as an origin. 

JEcliptic Limits. 

§ 474. Hence, if S M x be drawn perpendicular to the relative orbit, 
the length of S M x will measure the nearest approach of the moon's cen- 
tre to the axis of the earth's shadow. Make 

m = moon's hourly motion in longitude ; 
n = sun's " " " 

g = moon's hourly motion in latitude ; 
t = time from node to opposition or conjunction ; 
<p = M N S = angle which the relative orbit makes with the ecliptic. 

MS 
tan <p = 



NS' 



but supposing the motion uniform, 
MS = gt; 

N = mt\ S — nt\ 
.'. N S = m t — n t = (m — n) t ; 
9 



also 
and 

whence 



tan 9 = 



m — n 



The angle 9 is then known. 

§ 475. Denoting S M x by d, we have 



SN = 


J 

sin <p " 

NO 

SN 

ilue of 
== 4 . 


m 


(m — n 
9 


Y+ff* 


, substituting the vi 


m — 
SN, 
m 




NO 


^{m 


-nf + ff 2 
9 



(156) 



(157) 



(158) 



ECLIPSES. 



125 



Making 4 equal to that given in Eq. (149), the moon will just touch the 
earth's shadow, and N will become what is called the ecliptic limit ; 
that is, the least difference of longitude that can exist between the moon, 
and her nearest node at full, to avoid an eclipse of the moon. 

Taking the greatest value for 4, NO is found to be 12° 24', and least 
value it is found to be 9°. The first is called the greatest and the second 
the least lunar ecliptic limit. If therefore at the time of full moon the 
difference between the longitude of the moon and her nearest node exceed 
12° 24', there cannot be an eclipse; if less than 9°, there must be one; 
if less than 12° 24' and greater than 9°, there may or may not, depend- 
ing upon the inclination of the relative orbit and actual value of 4. To 
solve the doubt, we have the given difference of longitude between 12° 24' 
and 9°, and the inclination <p, to find S M x% If this latter be greater than 
J, there can be no eclipse ; if less, there must be one. 

§ 476. Again, making <4 equal to that given in Eq. (155), and pro- 
ceeding exactly as above, we find the greater and lesser solar ecliptic lim- 
its. The first is 18° 36' and the latter 15° 25'. 

Number of Eclipses. 

§ 477. Let NHNR' be the ecliptic, 
N and N' the moon's nodes. Take 
N L x , NL 2 , N'L Z , and N'L A , each equal 
to 18°.6, the greatest ecliptic limit. Then 
will L { L 2 and L 3 L A be each equal to 
37°.2, and the number of new moons 
that can happen while the sun is appa- 
rently describing these arcs, will deter- 
mine the number of solar eclipses that 
<jan occur in a single year. 

The mean daily motion of the moon's 
node is — 0°. 055 ; the mean apparent daily motion of the sun is 0°.985, 
and hence the apparent relative motion of the sun and node is 0°.985 — 
(— 0°.055) = 1°.04. A lunation is 29.53 days; and 29.53 X 1°.04 = 
30°.7ll2, say 30°. 71, is the mean motion of the sun from the node in a 
lunation. Omitting the proper motion of the vernal equinox as insignifi- 
cant in this estimate, its relative motion from the node in a lunation is 
29.53 days x 0°.055 = 1°.6241. These motions are incommensurable 
with each other and with 360°; and the vernal equinox, the node and 
sun with the moon in conjunction, will, in process of time, have any as- 
sumed positions with respect to each other at the beginning of the year. 




126 SPHERICAL ASTRONOMY. 

Taking the sun at M 9 , one degree to the east of L x (with moon in con- 
junction), will give one solar eclipse; and 37°. 2 — 1° = 36°. 2 l^ing 
greater than the arc described by the sun in a lunation, there will, at tli€ 
end of the first lunation, be another solar eclipse between N and L^ 
At the end of the sixth lunation, the sun will be at M^ in advance of 
L s by a distance equal to 30°. 71 X 6 — 179° = 5°.26, where there will 
be a third solar eclipse ; and 37°.2 — 5°.26 == 31°.84 being greater than 
arc described in a lunation, there must be a fourth solar eclipse before 
the sun passes Z 4 . At the end of the twelfth lunation, the suu will be 
30°.7l X 12 — 360° = 8°.52 to the east of Jf 2 , his initial place, where 
there will be a fifth solar eclipse, and this will be the last within the 
year, which will end 10.89 days after, this being the excess of the year 
over twelve lunations. 

Again, 18°. 6 — 1° = 17°. 6; and as in a semi-lunation the sun will 
pass over 15°. 35 of this arc, he will come to the distance 17°.G — 
15°.35 = 2°. 25 from the node iV 3 , and there will be a first lunar eclipse 
at the opposite node N^ When the moon was new at M z , tlie sun was 
18°.60 — 5°.36 = 13°.34 from the node N, and in half a lunation aftei 
will be 15°.35 — 13°.34 = 2°.01 beyond it, and there will be a second 
lunar eclipse, and no more within the year, for the next lunation will 
carry the sun beyond the lunar ecliptic limits. Had the initial place of 
the sun been taken at M v at a distance 4°. 26 to the west of the node 
N, at the beginning of the year, and the moon in opposition, it might 
have been shown that there would have been four eclipses of the sun 
and three of the moon. 

§ 478. The least solar ecliptic limit being 15°. 42, the arc L x Z 2 must 
be at least 30°. 84 ; which being greater than the arc passed over by the 
sun in a lunation, there must be at least one solar eclipse in each of thfc 
arcs L x L^ and L 3 X 4 , so that there must always be at least two eclipses 
of the sun in each year. 

The sun is less than a lunation in passing through the lunar ecliptic 
limits, and there may, therefore, be no eclipse of the moon within the year. 

To sum up, then, there may be seven eclipses within the year, and there 
may be only two. In the former case, five may be of the sun and two of the 
moon, or four of the sun and three of the moon ; and iu the latter, both 
must be of the sun. 

The Saros. 

§ 479. The synodic period of the moon is 29.53058, and that of the 
moon's node 346.6196 days. These numbers are to one another as 19 to 
223 nearly. If, therefore, the moon and her nodes be in syzyoy at the 



\ 






\S 



■. 



\ 






Plate 71. 




TO FRONT PAGE I'd / 







'tZ^S- 



CONSTI T U T I N V T II E M 



12? 




same time, they will be so again after 19 revolutions of the node, or 223 
lunations; so that the eclipses will recur again very nearly in the same 
order within the same period, which is about 18.027 years. This period 
is known as the Chaldean Saros. There are generally 10 eclipses in the 
saros, j}£,which 29 are lunar and 41 solar. y 

PHYSICAL CONSTITUTION OF THE MOX)K 

§ 480. Telescopes disclose certain varieties of illumination Ot the 
moon's surface, which can only arise from mountains and valleys. The 
shadows cast by the former lie in directions and are of lengths re- 
quired by the inclination of the solar rays to that portion of the moon's 
surface on which the mountains stand. The convex outline of the moon 
turned towards the sun is always circular and nearly smooth ; but the op- 
posite or elliptical border of the illuminated part is extremely ragged, and 
indented with deep recesses and prominent points. To places along this 
line the sun is just rising, and the neighboring mountains cast long black 
shadows on the plains below. As the sun rises these shadows shorter. ; 
and at full moon, when the solar light penetrates the mountain valleys and 
shines on every point of the field of view, no shadows are seen. 

§ 481. The summits of the lunar mountains often appear as small 
* bright points, or islands of light, beyond the edge of the illuminated part, 
as they catch the sunbeams before the intervening plains. As the sun ad- 
vances in altitude, these luminous patches expand, and finally unite with 
the general illumination, and the mountains appear as projections from its 
elliptical border. 

§ 482. To compute the Fi s- 91 - 

height of a lunar mountain, 
let E, M, and S be the cen- 
tres of the earth, moon, and 
sun respectively ; AC B D 
and DOC sections of the 
general surface of the moon 
by planes respectively perpen- 
dicular to EM and MS; 
then will the visible illumi- 
nated part of the disk be 

contained between CBD and the projection of D C on the section 
AC B D. Also let m be the top of a mountain just catching the solar 
rays that graze the general surface of the moon at ; B F the arc of a 




128 



SPHERICAL ASTRONOMY. 



great circle of this surface, 
and of which the plane passes 
through the top of the moun- 
tain and centres of the sun 
and moon ; n the point in 
which this arc is cut by 
the line Mm drawn from 
the top of the mountain to 
the moon's centre. 

Make 

r = Mn = radius of the moon ; 

s = FCO = JSMA = exterior angle of elongation ; 

(/ = Om — distance of in from ; 

x = mn = height of mountain ; 

a = the projection of y on the plane A C B D. 

Then, since the ray S' m is perpendicular to the section D C, it is in- 
clined to the section A C B D, under an angle equal to the complement 
of F C = 90° — s ; and we have 




a = y . cos (90° — s) = y 



whence 



Also 



a 

sin £* 



. i 



y 



o^r\ 



and, therefore, 



y= V#(2r + ;r); 
V ^ (2 r + #) 



a 

sin £ 



neglecting g in comparison with 2 r, we have 



2 r ' sin 2 £ 



a 2 

= — . cosec J € 
2r 



(159) 



V 



a being the observed distance of the bright spot from the boundary of the 
illumination, may be measured by means of the micrometer. 

§ 483. The heights of many of the lunar mountains have been thus 
computed, and they range through all elevations up to 23,000 English feet, 

§ 484. The lunar mountains are strikingly uniform in aspect. They 
are very numerous, especially towards the southern border, occupying by 
far the larger portion of the surface. They present almost universally a 
circular or cup-shaped form in ground plan, which becomes foreshortened 
into an ellipse towards the limb. The larger of these cups have for the most 




h& 



CONSTITUTION OF THE MOON 



129 



part flat bottoms, from each of which rises centrally a small, steep, conical 
hill, presenting in all respects the true volcanic character as exhibited by 

v Fig. 92. 




like districts on the earth, but with this peculiarity, viz. : that the bottoms 
are so deep as to lie below the geneiai surface of the moon, the internal 
depth beiug often twice or thrice the external height. 

§ 485. The heights of mountains in the immediate vicinity of each 
other being proportional to the length of their respective shadows, the 
depths of the pits or craters are easily computed from the heights of the 
edges above the general level, and the lengths of the shadows they cast 
internally and externally. 

§ 486. Through the Rosse telescope, the flat bottom of the crater 
called Albategnius, is seen to be strewed with blocks not visible through 
inferior instruments ; and the exterior of another, called AriMillus, is 
hatched over with deep gullies, radiating from a centre. 

§ 487. There are also extensive tracts of the lunar surface which are 
perfectly level, and present decided indications of an alluvial character, 
and yet there is a total absence of all appearances of deep water. 

§ 488. There are no clouds, or other indications of an atmosphere. 

A lunar atmosphere of a mean density equal to 1980th that of the 
earth, would give a horizontal refraction of 1", and cause the diameter of 
the moon, measured with a micrometer and estimated by the interval of 
a stars disappearance in an occupation, to differ ; would cause the limb of 
the moon, during a solar eclipse, to appear beyond the cusps externally to 
the sun's disk as a narrow line of light, extending for some distance along 
the edge ; and would extinguish very faint stars before occupations. But 
none of these phenomena are seen. During the continuance of a total 
luuar eclipse, when the light of the moon is so deadened as not to obliter- 
ate by contrast the feeble light of the smaller stars, the latter are seen to 
come up to tlie moon's limb and undergo sudden extinction, without any 
apparent displacement. 

§ 489. The light from the moon developes but feeble heat, for even 

9 



130 



SPHERICAL ASTRONOMY. 



when collected into the foci of large reflectors, it affects but little the 
thermometer; and there are no appearances indicating the slightest 
change of surface, such as would result from the periodical growth and 
decay of vegetation which accompany a change of seasons, 

§ 490. To an inhabitant of the moon, if there be such a thing, the 
earth must present the appearance of a moon 2° in diameter, exhibiting 
phases complementary to those the moon presents to us, but fixed in the 
sky, while the stars seem to pass slowly beside and behind it. It must 
appear clouded with variable spots, and belted with zones corresponding to 
our trade-winds. During a solar eclipse our atmosphere will appear as a 
naiTow, bright ring, of a ruddy color where it rests on the earth, gradually 
passing into faint blue, encircling the whole or part of the earth's disk. 



SATELLITES OF JUPITER. 

§ 491. The satellites of Jupiter, four in number, revolve about their pri- 
mary from west to east in planes nearly coincident with that of the planet's 
equator, and but slightly inclined to the ecliptic. 

§ 492. Their orbits appear, therefore, projected very nearly into straight 
lines, in which they oscillate to and fro, sometimes passing between the 
sun and Jupiter, causing an eclipse of the sun to the latter, sometimes en- 
tering the planet's shadow and being themselves eclipsed, and sometimes 
disappearing either behind the body of Jupiter or in transiting his disk. 

§ 493. Thus, let S be the sun; £}, the earth, of which the orbit is 
J$FGII\ J, Jupiter ; and efa b, the orbit of a satellite. The cone of Ju- 
piter's shadow will have its vertex at X, far beyond the orbit of the satel- 

Fig. 93. 




SATELLITES OF JUPITER. 131 

lite, and the penumbra, owing to the great distance of the sun and conse- 
quent sraallness of the angle at Jupiter subtended by his disk, will extend 
but little beyond the shadow within the limits of the satellite's oibit 
The satellite revolving from west to east, will cast a shadow upon Jupiter 
while passing from m to », will transit his disk from e to /, enter his 
shadow at «, emerge from it at b, and disappear behind the body of the 
planet while passing from c to d. 

§ 494. The shadows of the satellites are frequently seen crossing the 
disk of Jupiter. While in the act of transiting, the satellite generally dis- 
appears, its light being confounded with that of the planet, unless it hap- 
pens to be projected upon a dark belt, in which case it is visible. Under 
these circumstances it occasionally appears as a daik spot smaller than its 
shadow, which has led to the conclusion that certain of the satellites have 
now and then on their own bodies, or within their atmospheres, obscure 
spots of great extent. 

§ 495. From the eclipses of the satellites are obtained all the data for 
the determination of the laws of their motions. These eclipses are in gen- 
eral analogous to those of the moon, but in their details they differ con- 
siderably. The great distance of Jupiter fiom the sun and his great size, 
make his shadow much larger and longer than that 'of the earth. The sat- 
ellites are much smaller in proportion to their primary, and their orbits less 
inclined to his ecliptic, than in the case of the moon. From these causes 
the three interior satellites enter the shadow at every revolution, and are 
totally eclipsed; and although the fourth, fiom the greater inclination and 
distance of its orbit, sometimes escapes eclipse, yet it does so seldom. 

§ 496. Besides, these eclipses are not seen by us from the centre of mo- 
tion, as are those of the moon, but from some remote station, of which the 
place with respect to the shadow is ever changing. And while this cir- 
cumstance makes no difference in the time of the eclipses, it yet affects 
mateiially the visibility and the apparent relative situations of the planet 
and satellites at the instant of the latter's entering and quitting the shadow 

§ 497. A satellite never enters the shadow suddenly because of its sen 
sible diameter, and the time from the first perceptible loss of light to its 
total extinction will be that required by the satellite to describe about Ju- 
piter an angle equal to its apparent diameter as seen from the planet's cen- 
tre. The same is true of the emergence. Owing to the difference in tel- 
escopes and eyes, this becomes a source of discrepancy in the times assigned 
by different observers for the beginning and ending of an eclipse. But if 
both the immersion and emersion be observed by the same person and with 
the same telescope, the half sum of the two times, as given by a properly 



132 



SPHERICAL ASTRONOMY 



regulated time-keeper, will be that of apparent opposition measurably free 
from error. 

§ 498. The intervals between the oppositions give the synodic period, 
which, in Eq. (146), will give the mean motion, knowing that of Jupiter, 
and hence the sidereal period. Eq. (142). 

The satellites are named first, second, third, and fourth, according to 
their order of distance from Jupiter. 

The elements of the satellites' orbits will be found in the following 



Table. 



Sat. 


Sidereal period. 


Inclination of 
Mean orbit to a 
distance. ! fixed plane 

Rad.ofJ=l.P r °P ertoeach 


Inclination -r, t A 

of the Retrograde 


Mass: 
that of Jupiter 

1,000,000,000. 


1st. 
2d. 
3d. 
4th. 


d. h. m. s. 
1 18 27 33.506 

3 13 14 36.393 

7 03 42 33.362 

16 16 31 49.702 


6.04853 

9.62347 

15.35024 

26.99835 


O ' " 



27 50 
12 20 

14 58 


O ' " 

6 
1 5 
5 2 
24 4 


Tears. 

29.9142 
141.7390 
531.0000 


17328 
23235 

88497 
42659 



It will assist in forming some idea of the relative dimensions of Jupitei 
and his satellites to examine the following 

Table. 





Mean apparent 

diameter as seen 

from earth. 


Mean apparent 

diameter as seen 

from Jupiter. 


Diameter 

in 

miles. 


Mass. 


Jupiter. 


38.327 


' " 


87000 


1.0000000 


1st sat. 


1.017 


33 11 


2508 


0.0000173 


2d » 


0.911 


17 35 


2068 


0.0000232 


3d " 


1.488 


18 00 


3377 


0.0000885 


4th " 


1.273 


8 46 


2890 


0.0000427 



From which it follows that the first satellite appears to a spectator on 
Jupiter as large as our moon to us ; the second and third nearly equal to 
each other, and somewhat more than half the size of the first ; and the fourth 
about a quarter of that size. They frequently eclipse each other. The 
apparent diameters of the planet as seen from the satellites are 19° 49'; 
12° 29'; 1° 47/; 4° 25'. 

§ 499. Figure 93 shows that the eclipses take place to the west of Ju- 
piter, while the latter is moving from conjunction to opposition, and to the 



SATELLITES OF JUPITEE. 133 

east from opposition to conjunction. As Jupiter approaches to opposition, 
the line of sight from the earth becomes more nearly coincident with the 
direction of the shadow, and the place of the eclipse will be nearer and 
nearer to the body of the planet. When the earth comes to F, from which 
a line drawn tangent to the body of the planet will pass through &, the 
emersion will cease to be visible, and will, up to the time of opposition, 
take place behind the planet. Similarly, from opposition up to the time 
when the earth arrives at K, the immersion will be concealed from view. 
These remarks apply particularly to the third and fourth satellites, the 
proximity of the others to the planet being so great as to make it impossi- 
ble ever to see the immersion and emersion both at the same ec tpse. 

§ 500. The mean motions of the satellites are connected wy this re- 
markable law, viz.: If the mean angular velocity of the first fitellite be 
added to twice that of the third, the sum will equal three times nat of the 
second. If, therefore, from the mean longitude of the first sr ;ellite, in- 
creased by twice that of the third, three times the mean longitr \e of the 
second be subtracted, the remainder will be a constant quantity and this 
constant is found to be equal to 180°. This Laplace has showi. to be a 
consequence of the mutual attractions of the satellites for on< another. 
The first three satellites cannot, therefore, be eclipsed at the same ime. 

§ 501. While, however, the satellites cannot all be eclipsed at once, 
they may be, and, indeed, occasionally are, all invisible by the sinu-taneous 
eclipse of some, occultations of others, and transits of the rest. 

§ 502. The orbits of the satellites are but slightly eccentric, thi two in- 
ferior ones not at all so, so far as observation is capable of revealing eccen- 
tricity. Their mutual attractions produce iu them perturbations an .logous 
to those of the planets about the sun. These are investigated in physical 
astronomy. 

§ 503. By careful observations the satellites are found to exhibit i. o.rked 
fluctuations in respect to brightness. These fluctuations happen p tiodi- 
cally, and appear connected with the position of the satellites with nsMpect 
to the sun ; from which it is interred that they revolve upon their axe° "^ke 
our moon, each once in its sidereal period. 

§ 504. At one time the eclipses of Jupiter's satellites were much i ed 
in the determination of terrestrial longitude, but more modern metli. is, 
free from the objections referred to in § 497, have in a measure supplair "*1 
them. 



[34: SPHERICAL ASTRONOMi. 

Progressive Motion of Light. 

§ 505. 1 , these eclipses science is indebted for the discovery of the suc- 
cessive propagation and velocity of light. 

The earth's orbit being concentric with that of Jupiter and interior to it, 
the distance of these bodies is continually varying, the variation extending 
from the sum to the difference of the radii of the two orbits, making the 
excess of the greatest over the least distance equal to the diameter of the 
earth's oibit. Now, it was observed by Roemer, a Danish astronomer, on 
comparing together the eclipses during many successive years, that those 
which took place about opposition were observed earlier, and those about 
conjunction later than an average or mean time of occurrence. And con- 
necting the observed acceleration in the one case and retardation in the 
other with the variation of Jupiter's distance below and above its average 
value, he found the difference fully and accurately accounted for by allow- 
ing 16 m 26 5 .6 for light to traverse the diameter of the earth's orbit. In 
other words, using the figure of a cord moving in the direction of its length 
from the satellite to the earth to illustrate the flow of luminous waves in 
the same direction, if the cord were severed at the edge of Jupiter's shadow, 
the severed end would be 16 m 26 s .6 longer in reaching the earth when the 
planet is in conjunction than in opposition, having a greater distance to 
travel in the first case by the diameter of the earth's oibit = 190,000,000 
miles, than in the second. The satellite is seen long after it has entered 
the shadow, and is invisible long after it has emerged from it. Dividing 
the diameter of the earth's orbit by 16 m 26\6 reduced to seconds, the ve- 
locity of light is found to be 192,000 miles a second. 

SATELLITES OF SATURN. 

§ 506. Eight satellites are known to accompany Saturn. They revolve 
about him from west to east, and in planes nearly coincident with that of 
the planet's ring, except the eighth, whose orbit is inclined to this latter 
plane under an angle of about 12° 14'. This satellite is also distinguished 
from the others by its remoteness from the planet, its distance being 
2.3 times that of the most distant of the others, and equal to 64 times 
the equatorial radius of Saturn, resembling in this respect our own moon. 
It is also remarkable for the exhibition of greater variety of illumination 
in different parts of its orbit than any other known secondary. Indeed, so 
feeble is the light which it reflects to the earth when to the east of Saturn 
that it becomes invisible through ordinary telescopes ; and from this defi- 



SATELLITES OF SATURN. 



135 



ciency of light occurring constantly on the same side of Saturn, as seen 
from the earth, it is inferred that this satellite revolves on its axis once 
during its sidereal period. 

§ 507. The next in order, proceeding inwardly, is so obscure as to have 
eluded the observations of astronomers until very recently. It was dis- 
covered simultaneously by Mr. BoDd, of Cambridge, U. S., and Mr. Lassell, 
of Liverpool, England, in 1848. 

§ 508. The next in order, proceeding in the same direction, is by far 
the largest and most conspicuous of all, and probably not inferior to Mars 
in size. 

§ 509. The next three in order are very small, and require pretty pow- 
erful telescopes to see them, while the two interior, which just skirt the 
edge of the ring, can only be seen with telescopes of extraordinary power 
and perfection, and under the most favorable atmospheric circumstances. 
When first discovered, they appeared to thread the excessively thin film 
of light reflected from the edge of the ring then turned towards the earth, 
and for a short time to advance off at either end, speedily to return again. 

§ 510. Owing to the obliquity of their orbits to the plane of Saturn's 
ecliptic, there are no eclipses, occultations, or transits of the satellites, or 
shadows on the disk of the primary, except at the time when the ring is 
seen edgewise, and their observation is attended with too much difficulty 
to be of any practical use, like the corresponding phenomena of Jupiter's 
satellites, for the determination of terrestrial longitude. 

§ 511. The names and elements of Saturn's satellites are given in the 

following 

Table. 



Names and 
Order of 
Satellites. 


Sidereal Period. 


Mean 
Distance. 


Epoch 

of Ele- 
ments. 


Mean Longi- 
tude at the 
Epoch. 


Eocentri- 
city. 


Perisatur- 
num. 


1. Mimas . . . 


d. h. m. s. 
22 37 22.9 


3.3607 


1790.0 


O ' " 

256 58 48 






, 2. Eneeladus 


1 08 53 06.7 


4.3125 


1836.0 


67 41 36 






3. Tetbys... 


1 21 18 25.7 


5.3396 


'« 


313 43 48 


0.04? 


54° ? 


4. Dione. . . . 


2 17 41 08.9 


6.8398 


<« 


327 40 48 


0.02 ? 


42 ? 


5. Rhea 


4 12 25 10.8 


9.5528 


<( 


353 44 00 


02? 


95 \ 


! 6. Titan 


15 22 41 25.2 


22.1450 


1830.0 137 21 24 


0.029314 


256° 38M1 


| 7. Hyperion. 


22 12 ? ? 


28. ± 








8. Iapetus . . 


79 07 53 40.4' 


64.3590 


1790.01269 37 48 




i 



The longitudes are reckoned in the plane of the ring from its descend- 
ing node on the ecliptic. The apsides of Titan have a direct motion of 
30' 28" per annum in longitude on the ecliptic. 



136 



SPHERICAL ASTRONOMY. 



§ 512. The periodic times of the first four satellites in order of distance 
from Saturn are connected by this law, viz. : The period of the third is 
double that of the first, and the period of the fourth is double that of the 
second ; the coincidence being exact to within ¥ J ^ part of the larger 
period. 

SATELLITES OF URANUS. 

§ 513. Uranus is believed to have six satellites, which revolve about the 
primary from east to west, in orbits nearly, if not quite, circular, and which 
make with the ecliptic an angle of 78° 58'. They thus differ from all the 
other known bodies of the solar system both in the direction of their mo- 
tion and inclination of their orbits, which latter, as well as the places of 
the nodes, have undergone no sensible change, during at least one-half of 
the planet's period around the sun. 

The elements of these satellites, as far as known, are given in this 

Table. 



1 

Sat. Sidereal Eevolution. 


Mean 
Distance. 


Epoch of passing Ascend- 
ing Node. 


Nodes and Inclination. 








Gr. T. 




1 


4 J ? h 






Inclination of orbits to 


2 


8 16 h 56™3H3 


17 


1787. Feb. 16, 0* 10 m 


the ecliptic, 78° 58' ; 
ascending node in 
longitude, 165° 30'. 


! 3 


10 23 ? 


19 8? 




4 


13 11 07 12.6 


22 8 


1787. Jan. 1, h 28 m 


(Equinox of 1798.) 


5 


38 2 ? 


45 5? 




Motion retrograde, 


6 


107 12 1 


91 0? 




and orbits nearly 
circular. 



§ 514. The satellites of Uranus require very powerful and perfect tele- 
scopes for their observation. The second and fourth are far the most con- 
spicuous, and their periods and distance have been ascertained with toler- 
able certainty. The first and third have also been observed since their 
original announcement, but of the existence of the fifth and sixth we have 
not the same evidence. Sir John* Herschel is of opinion that if future 
observations should assign them places, they would be exterior to that of 
the fourth. 

515. When the earth is in the plane of the orbits or nearly so, the ap- 
parent paths of the satellites are straight lines or very elongated ellipses, 
in which case these secondaries become invisible long before they come 
up to the disk of the planet, in consequence of the superior light of the 
latter, so that it is not possible to observe their occultations, eclipses, ant? 
transits. 



COMETS. 137 



SATELLITES OF NEPTUNE. 



§ 5Z6. If the observation of the satellites of Uranus be difficult, those 
of Neptune, owing to the great distance of this planet, must offer still 
greater difficulties. Of the existence of one satellite there remains no 
doubt. Its sidereal period about the planet is nearly 5.9 days ; its mean 
distance is fourteen times Neptune's semi-diameter; and its orbit is in- 
clined to the plane of the ecliptic under an angle of about 35°. 



COMETS. 

§ 517. Comets differ from all the primary bodies with which we have 
thus far been concerned, in their appearance, the shape and inclination of 
their orbits, and in following no rule, as a class, with regard to the direc- 
tion of their motions. They are of various sizes, some being visible to the 
naked eye even in daytime, while others require the aid of telescopes even 
at night to see them. 

§ 518. The larger consist for the most part of an ill-defined mass, called 
the head, from which, in a direction opposite the sun, proceeds a train, of 
greater or less extent, called the tail. 

Fi» 94. 





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§ 519. The head is much brighter towards its centre. Sometimes this 
increase of illumination terminates in a bright spot, called a nucleus, the 
surrounding haze which makes up the rest of the head being called the 
coma. 

§ 520. The tail appears to consist of two streams of luminous matter 
which, starting from a point near the head, and on the side towards the 
sun, pass suddenly to the opposite side, and grow broader and more dif- 
fused as they increase in length ; they commonly unite at a little distance 
from the head, but sometimes continue distinct for the greater part of their 
course. This appendage has been known to attain the enormous length 
of forty-one millions of miles, and to stretch over 104 degrees of the celes- 
tial sphere. 

§ 521. The tail is not, however, an invariable appendage of comets, 



|38 SPHERICAL ASTRONOMY. 

mauv of the brightest having "been seen with little or none, and others as 
round and well-defined as Jupiter. 

§ 522. On the other hand, there are instances of comets with many 
tails or streamers, spreading out like an immense fan, and extending to the 
distance of some 30 degrees of the celestial vault. One is recorded as 
having two tails, making with each other an angle of 160°, the fainter 
being tinned towards, the other from the sun. 

The tails are often curved, bending, in general, towards that part of 
space which the comet has left, as if retarded by the opposition of some 
resisting medium. 

§ 523. The smaller comets, such as are only visible through telescopes, 
and which are by far the most numerous, present no appearance of a tail, 
and seem as round or oval vaporous masses, more luminous towards the 
centre, where, in some instances, a small stellar point has been seen, but 
without auy distinct nucleus or other signs of a solid body. Stars of the 
smallest magnitude, such as would be obliterated by a moderate fog. are 
seen through their brightest part. 

§ 524. A comet never exhibits the least signs of phases; but, on the 
contrary, appears as a mass of thin vapor, either self-luminous, or easily 
penetrated by the luminous waves from the sun, which are reflected from 
its interior parts as from its exterior surface. 

§ 525. The tail, where it comes up and surrounds the head, is yet sep- 
arate from the latter by an interval less luminous, as if sustained and kept 
from contact by a transparent stratum of atmosphere; and seems to be a 
kind of hollow envelope of a parabolic form, inclosing the head near its 
vertex. 

§ 52G. The number of recorded comets is very great, amounting to sev- 
eral hundred ; and when it is considered that in the earlier stages of as- 
tronomy, before the invention of the telescope, only large and conspicuous 
ones could be noticed, and that, since due attention has been paid to the 
subject, scarcely a year passes without the observation of one or two of 
these bodies, and sometimes two or three have appealed at once, it may 
very reasonably be supposed that many thousands exist "Multitudes must 
escape observation by reason of their paths traversing only that part of the 
heavens which is above the horizon in daytime. Comets so circumstanced 
can only become visible during a total eclipse of the sun — a coincidence 
which is related to have taken place sixty years before Christ, when a 
a large comet was observed near the sun. 

§ 527. The motion of comets is characterized by the greatest irregular^ 
ity. Sometimes they appear in sight for a few days only, at others for 



COMETS. 139 

many months. Some move very slowly, others with vast velocity ; and 
not unfrequently the two extremes of speed are exhibited by the same in- 
dividual in different parts of its path. Some pursue a direct, others a ret- 
rograde, and others a tortuous and very irregular course ; nor are they 
confined, like the planets, to any particular region of the heavens, but 
traverse indifferently every part alike. 

§ 528. Their variations in apparent size, while visible, are equally re- 
markable ; sometimes they make their appearance as faint, slow 7 -moving 
objects, with little or no tail; by degrees they accelerate their speed, en- 
large and extend their tail, which increases in length and brightness till 
they approach the sun near enough to be lost in his light. After a time 
they again emerge on the opposite side, receding from the sun. It is now 
for the most part they shine forth in all their splendor, and display their 
tails in greatest length and development. As they continue to recede 
from the sun their motion diminishes, their tails subside about the head, 
which glows continually feebler till lost in the distance, from which by far 
the greater number have never returned ; thus indicating their paths to be 
along the parabola or hyperbola. 

§ 529. These seemingly irregular and capricious movements are fully 
explained by the doctrine of universal gravitation, and are no other than 
consequences of the laws of elliptic, parabolic, or hyperbolic motions. But 
the physical changes of the head, the process by which it builds up the 
enormous tail, takes it down again, and wraps it as a mantle about itself; 
the position of the tail as regards the direction of the sun, the multiplicity 
of tails, and other physical phenomena to be noticed presently, remain 
without satisfactory solution. 

§ 530. The elements of a comet's orbit are readily computed from three 
observed places, exactly as in the case of a planet ; and the comet usually 
takes the name of the computor who thus first defines its track through 
the heaveus. 

The elements of a few now reckoned among the permant nt members of 
the solar system, will be found in the following table : 



HO 



SPHERICAL ASTRONOMY 





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COMETS. 14J 

§ 531. By far the most interesting of these comets is that of Halley. 
Its last return took place according to prediction in 1835. While yet 
remote from the sun in its approach to that luminary, its appearance was 
that of an oval nebula without tail, and having a minute point of concen- 
trated light eccentrically situated within. Soon its tail began to be devel- 
oped, and increased rapidly till it reached its greatest length, about 20 
degrees, when it decreased with such haste as to disappear entirely before 
perihelion passage. When the ail first began to form, the nucleus be- 
came much brighter, and threw out a jet or stream of light towards the 
sun. This ejection continued, with occasional intermission, as long as the 
tail continued visible. Both the form and direction of this luminous 
stream underwent singular and capricious alterations, the different phases- 
succeeding one another with such rapidity that no two successive nights 
presented the same appearance. At one time the jet was single, at others 
fan-shaped, while at others two, three, or more jets were darted forth in 
different directions, the principal one oscillating to and fro on either side 
of the line drawn to the sun. These jets, though very bright at their 
point of emanation from the nucleus, faded away, and became diffused as 
they expanded into the coma, at the same time curving backward as if 
thrown against a resisting medium. After its perihelion passage, the 
comet was not seen for two months, and at its reappearance presented 
itself under a new aspect. There was no longer a vestige of tail ; it 
seemed to the naked eye a hazy star of the fourth magnitude, and through 
a powerful telescope a small round well-defined disk, rather more than 2' 
in diameter, surrounded by a nebulous coma of much greater extent. 
Within the disk, and somewhat removed from its centre, appeared a mi- 
nute but bright nucleus, from which extended, in a direction opposite the 
sun, a short vivid luminous ray. As the comet receded from the sun, the 
coma disappeared, as if absorbed into the disk, which increased so rap- 
idly as in one week to augment its volume in the ratio of 40 to 1. And 
so it continued to swell out, with undiminished rate, until from this cause 
alone it ceased to be visible, the illumination becoming fainter as the 
magnitude increased. While this increase of dimensions proceeded, the 
form of the disk passed, by gradual and successive additions to its length 
in the direction opposite to the sun, to that of a paraboloid, the side towards 
the sun preserving its planetary sharpness, but the base being so faint and 
ill-defined, as to indicate that if the process had been continued with suffi- 
cient light to render it visible, a tail would ultimately have been observed. 
The parabolic envelope finally disappeared, and the comet took its leave 
as it came — a small round nebula, with a bright point in or near the 



142 SPHERICAL ASTRONOMY. 

centre. Figures 5 to 10 inclusive, of plate, taken m order, show some of 
the successive aspects of this comet at its last appearance. 

§ 532. Many other great comets are recorded, all affording peculiarities 
more or less interesting. 

§ 533. On comparing the intervals between the successive returns of 
Encke's comet, its periods are found to be continually shortening ; that is, 
its mean distance from the sun, or semi-major axis of its orbit, diminishes 
by slow and regular degrees, and at the rate of about u .l 1 during each 
revolution. This is attributed to the resistance of the ethereal medium 
which fills the planetary space, and serves as the medium for the transmis- 
sion of light. This resistance checks the velocity, diminishes the centrifu- 
gal force, and skives to the sun more effect in drawing the comet towards 
itself. It will probably ultimately fall into that body. Like the comet of 
Hallev, its apparent diameter is found to diminish as it approaches to, and 
to increase as it recedes from the sun. It has no tail, and presents to the 
view only a small ill-defined nucleus, eccentrically situated within a more 
or less elongated oval mass of vapors, being nearest to that vertex which is 
towards the sun. 

§ 534. Biela's comet is scarcely visible to the naked eye ; its orbit 
nearly intersects that of the earth, and had the latter, at the time of its 
passage in 1832, been a month in advance of its actual place, it would 
have passed through the comet. 

At its last appearance it separated itself into two parts, which contin- 
ued to journey along together, side by side, through an arc of 70 degrees 
of their orbit, keeping all the while within the same field of view of a tel- 
escope directed towards them. Both had nuclei, both had short tails par- 
allel to one another, and perpendicular to their line of junction. At first 
the new comet was extremely small and faint in comparison with the old : 
the difference both in light and size diminished till they became equal ; 
after which the new comet gained the superiority of light, presenting, ac- 
cording to Lieut. Maury, the appearance of a diamond spark. The old 
comet soon, however, recovered its superiority, and the new one began to 
fade, till finally the comet was seen single before it disappeared. While 
this interchange of light was going on, the new comet threw out a faint 
bridge-like arch of light, which extended from one to the other. When 
the original comet recovered its superior brightness, it in its turn threw 
forth additional rays, so as to present the appearance of a comet with three 
tails, forming with one another angles of about 120°. The distance be- 
tween the comets at one time was about 39 times the equatorial radius of 
the earth, or less than two-thirds the distance of the moon from the earth. 











Plate, VII. 




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TO FUOWT PAGE 142 



COMETS. 1^3 

§ 535. The orbits of comets being very eccentric, and inclined under all 
sorts of angles to the ecliptic, these bodies must pass near to the planets, 
and be more or less affected by their disturbing action. 
' One passed Jupiter at the distance of -£ 6 - of the radius of that planet's 
orbit, and the earth, three years afterwards, at seven times the moon's dis- 
tance. This comet was found by Lexell to have passed its perihelion in 
an elliptical orbit, of which the eccentricity was 0.7858, and with a pe- 
riodic time of about five and a half years, having, in all probability, been 
drawn into this path by the perturbating action of Jupiter and the earth at 
its previous visits. Its next return could not be observed by reason of the 
relative places of its perihelion and of the earth, and before another revo- 
lution could be accomplished, it passed within the orbit of Jupiter's fourth 
satellite, and has never been seen since. The action of Jupiter doubtless 
changed its orbit into an extremely elongated ellipse, or perchance into a 
parabola or hyperbola ; and what is most remarkable, none of Jupiter's 
satellites suffered any perceptible derangement — a sufficient proof of th« 
small ness of the comet's mass. 

§ 536. The great number of comets which appear to move in para- 
bolic orbits, or elliptical orbits so elongated as not to be distinguished from 
them, has given rise to an impression that these bodies are extraneous to 
our system, and that our elliptic comets owe their permanent denizenship 
within the sphere of the sun's dominant attraction to the retarding action 
of one or other of the planets near which they may have passed, and by 
which their velocity was reduced to compatibility with elliptic motion. 
A similar disturbing cause, acting to increase the velocity, would give rise 
to a parabolic or hyperbolic orbit, so that it is not impossible for a comet 
to be drawn into our system, retained during many revolutions about the 
sun, and finally expelled from it, never more to return, as was probably 
the case with that of Lexell. 

§ 537. The fact that all the planets and nearly all the satellites move 
in one direction about the sun, while retrograde comets are very common, 
would go far to assign them an extraneous origin. From a consideration 
of all the cometary orbits known in the early part of the present century, 
Laplace found that the average situation of their planes was so nearly per- 
pendicular to the ecliptic as to afford no presumption of any cause biasing 
their inclinations. And yet as the planes of the elliptical orbits approach 
that of the ecliptic, the number of direct comets increases ; and a plane of 
motion coincident with that of the earth, and periodicity of return, are de- 
cidedly favorable to direct motion. 



-[44 SPHERICAL ASTRONOMY. 

STARS. 

§ 533. Besides the bodies composing the solar system, there are a 
countless multitude of others which, because they retain their relative places 
sensibly unchanged are called, though improperly, fixed stars. Like our 
sun they are poised in space, are self-luminous, and in all probability are 
centres of planetary systems. 

§ 539. Among these stars, which at first view seem scattered over the 
celestial vault. at random, appears, every evening, a bright band, called the 
milky ivay, that stretches from horizon to horizon and forms a zone com- 
pletely encircling the heavens. It divides in one part of its course into 
two branches, which unite again after remaining separate for 150° of their 
course. 

§ 540. The most refined observations have been able to assign to none 
of the stars a sensible geocentric, and to but very few only an exceeding 
small and uncertain annual parallax ; while the most powerful magnifiers 
have thus far failed to reveal an appreciable disk. 

But little can, therefore, be known of their distances, nothing at all of 
their real dimensions, and the only means by which one may be distin- 
guished from another are in the character and intensity of their illumina- 
tion. 

§ 541. It is usual to arrange the stars into classes called magnitudes, 
and this without reference to their location in the heavens. The brightest 
are said to be of the first magnitude, those which fall so far short of the 
first degree of brightness as to make a strongly marked distinction, are 
classed in the second, and so on down to the sixth or seventh, which com- 
prise the smallest stars visible to the naked eye in the clearest and darkest 
night. 

§ 542. Beyond this, however, telescopes continue the range of visibility 
down to the 16th ; nor does there seem any reason to assign a limit to the 
progression, for every increase in the dimensions and power of telescopes 
has brought into view multitudes innumerable of objects invisible before ; 
and, for any thing experience has taught us, the number of stars may, to 
our powers of enumeration, be regarded as absolutely without limit. 

§ 543. The mode of classification into orders is entirely arbitrary. Of 
a multitude of bright objects, differing in all probability intrinsically both 
in size and splendor and arranged at unequal distances, one must appear 
the brightest, another next below it, and so on. An order of succession 
must exist, and when it is gradual in degree and indefinite in extent, tc 
draw a line of demarkation is matter of pure convention. 



STARS. 



145 



§ 544. Sir Jolm Herschel proposes to make the scale of decreasing 
brightness of the stars which head the several ordeis of magnitudes, to 
vary inversely as the squares of the natural numbers, or as 1, i, ^-, j^, -^V? 
&c. ; that is, the brightest star Of the first magnitude shall be four times 
that of the brightest of the second, nine times that of the brightest of the 
third, and so on : stars of intermediate brightness to be expressed deci- 
mally. Thus a star half way in brightness between the brightest of the 

third and of the fourth magnitudes would be expressed by - a . 

On the hypothesis that all the stars possess the same intrinsic brightness, 
coupled with the fact that the distance of the same luminous object varies 
inversely as the square root of its apparent brightness, the mere mention of 
the magnitudes of the stars would suggest, according to this classification, 
their relative distribution through space. 

§ 545. To accomplish this F*& 96. 

photometrical classification, he 
proposes to receive the light from 
the planet Jupiter, at A, on the 
first face of a triangular prism, 
so as to fall on the second face 
at C under an angle of total 
reflection; this light, on its 
emergence from the third face, 
being received upon a convex 
lens J9, would form an image 
of Jupiter's disk at F. An eye 
placed at E, within the field of 
the diverging waves, would re- 
ceive the light from this image 
and that from a star proceeding 
along the line BE. The ap- 
parent brightness of Jupiter's 
image would vary inversely as 
the square of FE, because this 

planet has no sensible phases, and under the same atmospheric circum- 
stances is of a constant brightness, while that of the star would be constant 
for all positions of the eye, and by altering the place of the latter the star 
and the image may be made to appear equally bright. The value of EF 
being ascertained for different stars, their relative brightness becomes 
known. 

10 




146 



SPHERICAL ASTRONOMY. 



§ 546. Astronomers have generally agreed to restrict the first magni- 
tude to about 23 or 24 stars, the second to 50 or 60, the third to about 
200, and so on, their numbers increasing rapidly as we proceed in the order 
of decreasing brightness, the number of stars registered to include the sev 
enth magnitude being from 12 to 15 thousand. 

§ 547. Stars of the first three or four magnitudes are distributed pretty 
uniformly over the celestial sphere, the number being somewhat greater, 
however, especially in the southern hemisphere, along a zone following the 
course of a great circle through the stars called s Ononis and a Crusis. 
But when the whole number visible to the naked eye are considered, they 
increase greatly towards the borders of the milky way. And if the tele- 
scopic stars be included, they will be found crowded beyond imagination 
along the entire extent of that remarkable belt and its branches. Indeed, 
its whole light is composed of stars of every magnitude from such as are 
visible to the naked eye to the smallest point perceptible through the beM 
telescopes. 

§ 548. The general course of the milky way, neglecting occasional de- 
viations and followiug the greatest brightness, is that of a great circle in- 
clined to the equinoctial under an angle of 63°, and cutting that circle in 
right ascension h 47 m and 12 h 47 m , so that its northern and southern poles 
are respectively in right ascension 18 h 47 m and 6 h 47 m . 

§ 549. This great circle of the celestial sphere with which the general 
course of the milky way most nearly coincides, is called the gallactic circle. 
To count the number of stars of all magnitudes visible in a single field of 
a telescope, and to alter the field so as to take in successively the entire 
celestial sphere, is to gauge the heavens. 

§ 550. A comparison of many different gauges has given the average 
imrnber of stars in a single field of 15' diameter, within zones encircling 
the poles of the gallactic circle, found in the following 



Table. 



Zones of North Gallactic 
Polar distance. 


Average 
in 


Number oi Start 
field of 15'. 


0° 


to 


15° . 


. 


4.32 


15 


to 


30 


. 


5.42 


30 


to 


45 


. 


8.21 


45 


to 


60 


. . 


13.61 


60 


to 


75 




24.09 


75 


to 


90 




53.43 









STARS. 






Sones of South Gallactic 
Polar distance. 




Average Number of Stare 
iu field of 15'. 


0° 


to 


15° 


. 


. 


6.05 


15 


to 


30 


. 


. 


6.62 


SO 


to 


45 


• 


. 


9.08 


45 


to 


60 


. 


. 


13.49 


60 


to 


75 


. 


. 


26.29 


75 


to 


90 


. , 


. 


59.06 



147 



Fig. 96. 




§ 551, This shows that the stars of our firmament, instead of being 
scattered in all directions indifferently through space, form a stratum of 
which the thickness is small in comparison with its length and breadth, 
and that our sun occupies a place 
somewhere about the middle of 
the thickness, and near the point 
where it subdivides into two prin- 
cipal lamina?, inclined under a 
small angle to one another. For 
to an eye so situated, the apparent 
density of stars, supposing them 
pretty equally scattered through 

the space they occupy, would be least in the direction A S, perpendicular 
to the laminae, and greatest in that of its breadth SB, S (7, or $Z>; in- 
creasing rapidly in passing from one direction to the other. 

§ 552. For convenience of reference and of mapping, the stars are sep- 
arated into groups by conceiving inclosing lines drawn upon the celestial 
sphere after the manner of geographical boundaries on the earth. The 
groups of stars within such boundaries are called constellations. The 
brightest star in each constellation is designated by the first letter of the 
Greek alphabet, the next brightest by the second, and so on till this alpha- 
bet is exhausted, when recourse is had to the Roman alphabet, and then to 
numerals. A star will be known from the name of the constellation and 
the letter or numeral: thus, a Centauri, 61 Cygni. Many of the bright- 
est stars have also proper names, as Sirius, Arcturus, Pviaris, <fec. 

§ 553. If, in Eq. (28), p denote the radius of the earth's orbit, it become? 
the annual parallax, d the star's distance, and w as before the number of 
seconds in radius unity. That equation gives 

d - = "- (160) 

p v v / 

§ 554. A line connecting the earth and a star would in the course of a 
year describe the entire surface of a cone of which the vertex would be the 



148 SPHERICAL ASTRONOMY. 

star, and the base the orbit of the earth. The intersection of the nappe of 
this cone beyond the star with the celestial sphere would be an ellipse, and 
the apparent orbit of the star, arising from heliocentric parallax. The 
greater axis of this ellipse would be double the annual parallax. 

§ 555. The stars floating, as it were, in space, and being subjected to 
the laws of universal gravitation, must each have a proper motion. In con- 
sequence of their vast distances from one another this motion may be com- 
paratively slow, and their excessive distance from us almost conceals it, re- 
quiring years to describe spaces sufficiently great to subtend sensible angles 
at the earth. By comparing the relative places of stars at remote periods 
this proper motion has been detected and measured in a great many in- 
stances. 

§ 556. Stars having the greatest proper motion are inferred to be near- 
est to us, and this has determined the selection of certain stars in preference 
to others in the efforts which have been made to ascertain their paral- 
laxes. 

§ 557. Two methods have been pursued. First, to find by careful me- 
ridional observations of right ascensions and declinations, cleared from re- 
fraction, nutation, aberration, and proper motion, the places of the star 
throughout the year, and thence the distance between those places most 
remote from one another. This is double the annual parallax. 

Second, after selecting two stars very near to one another, and of which 
one has an obvious proper motion and the other not, to measure with the 
heliometer or micrometer their apparent distances apart, and to note the 
corresponding positions of the line joining them throughout the year ; then 
to construct therefrom, after correcting for proper motion, the annual path 
of the moving star. Its longer axis will be double the annual parallax. 
This second is greatly the preferable method. The stars being separated 
by a few seconds only, they will be equally affected by refraction, nutation, 
and aberration, none of these depending upon actual distance. The method 
supposes the apparently immovable star to be immensely distant beyond 
the movable one. 

By the first method Professor Henderson found the parallax of a Cen- 
tauii to be 0".913 ; and by the second M. Bessel that of 61 Cygni to be 
0".348. 

§ 558. Assuming the parallax of a Centauri = 1", to avoid multiplicity 
of figures, substituting it for <k in Eq. (160), and writing the numerical 
value of w, we have 

d oj 

- = - = 203265 . . . (161) 

p it ' 



STARS. 149 

and in this proportion at least must the distance of the fixed stars exceed 
the distance of the sun from the earth. 

Substituting for p its value, say in round numbers 95,000,000 of miles, 
and we have 

d = 206265 X 95000000 = 19595175000000" 1 , 

or about twenty billions of miles. 

§ 559. Denoting the velocity of light by v, the time required for it to 
traverse the distance which separates the star from the earth by t, we have 

first, § 505, 

v = 192000™, 
and 

d 

t = - = 3 y .23 ; 

v 

that is to say, it would require light three years and a quarter to come 
from the nearest fixed star to the earth. And as this is the inferior limit 
which it is already ascertained that even the brightest and therefore, in the 
absence of all other indications, the nearest stars exceed, what is to be al- 
lowed for the distances of those innumerable stars of the smaller magni- 
tudes which the most powerful telescopes disclose in the remote regions of 
the milky way ? 

§ 560. The space penetrating power of a telescope, or the comparative 
distance to which a star would require to be removed in order that it may 
appear of the same brightness through the telescope as it did before to the 
naked eye, may be calculated from the aperture of the telescope as com- 
pared with that of the pupil of the eye, and from its power of reflecting or 
of transmitting incident light. The space penetrating power of the tele- 
scope employed on the gauge stais referred to in § 550 was 75. A star 
of the 6th magnitude removed to 75 times its distance would therefore still 
be visible, as a star, through that instrument, and admitting such a star to 
have 100th part the light of a standard star of the 1st magnitude, it will 
follow, from the law of illumination and distance, that such standard star 
if removed 75 X 10 = 750 times its distance would excite in the eye, 
when viewed through the telescope, the same impression as a star of the 
6th magnitude does in the naked eye. Among the infinite number of 
stars in the remoter regions of the milky way it is but reasonable to con- 
clude that there are many individuals intrinsically as bright as those which 
immediately surround us. The light of such stars must, therefore, have 
occupied 750 X 3.25 = 2437.5 years in travelling over the distance 
which separates them from our own system. And it follows that when 
we observe the places and note the appearances of such stars, we are only 



150 



SPHERICAL ASTIJOKOMY, 



0.913; 


Henderson. 


0.348; 


Bessel. 


0.261 ; 


Struve. 


0.230; 


Henderson. 


0.226; 


Peters. 


0.133; 


u 


0.127; 


u 


0.067 ; 


u 


0.046 ; 


u 



reading their history more than two thousand years before. Nor is this 
conclusion, startling as it may appear, to be avoided without attributing 
an inferiority of intrinsic illumination to all the stars of the milky way — 
an alternative much less in harmony, as we shall see presently, with astro 
nomical facts connected with other sidereal systems, revealed by the tele- 
scope, than are the views just taken. 

§ 561. Of some of the stars whose parallaxes have been determined, 
the values of the parallaxes, and the names of the discoverers, are given in 
this 

Table. 

a Centau ri 
61 Cygni 
a Lyra . 

Sirius 
1831 Groombridge 
i Ursse Majoris 
Arcturus 
Polaris 
Capella 

§ 562. As remarked in the beginning of this chapter, the very best 
telescopes afford only negative information respecting the apparent diam- 
eters of the stars. The round and well-defined planetary disks which good 
telescopes exhibit are mere optical illusions, these disks diminishing more 
and more in proportion as the aperture and power of the instiument are 
increased. And the strongest evidence of a total absence of perceptible 
dimensions is the fact, that in occupations of the stars by the moon, the 
extinctions are absolutely instantaneous. 

If our sun were removed to the distance of a Centaury its apparent di- 
ameter of 32' 3" woild be reduced to only 0".0093, a quantity which no 
improvement of our present instruments can ever show with an apprecia- 
ble disk. 

§ 563. The star a Centauri has been directly compared with the moon 
by the method of § 545. By eleven such comparisons, after making due 
allowances for known sources of error, it was found that the light of the 
full moon exceeded that of the star in the proportion of 27408 to 1. 
Wollaston found the proportion of the sun's light to that of the moon to 
be as 801072 to 1. Combining these results, the light we receive from 
the sun is to that from a Centauri as 21,955,000,000. or about twenty- 
two thousand millions to one. Hence, the illumination being inversely as 



STARS. 151 

the sq lare of the distance, the intrinsic splendor of this star is to that of 
the sun as 2.3247 to 1. The light of Sirius is four times that of a Cen- 
taury and its parallax only 0".230, which give to Sirius a splendor equal 
to 140.2 times that of the sun. 

§ 564. Periodical Stars. — Many of the stars, which in other respects 
are no way distinguished from the rest, undergo periodical increase and 
diminution of brightness, involving in one or two instances complete ex- 
tinction and renovation. These are called periodical stars. 

§ 565. The most remarkable star in this respect is o Ceti, sometimes 
called Mira. It appears at variable intervals, of which the mean is 33 l d 
I5 h 1 m . It retains its greatest brightness for a fortnight, being on some 
occasions equal to a large star of the second magnitude ; decreases for 
about three months, becoming completely invisible to the naked eye for 
about five months, and increases for the remainder of the period. Such is 
the general course of its phases. It does not always return to the same 
degree of brightness, nor increase nor decrease by the same gradations, 
neither are the successive intervals of maxima equal. The mean interval 
is subject to a cyclical fluctuation embracing eighty-eight such intervals, 
and having the effect to shorten and lengthen the same about 25 days one 
way and the other. 

§ 566. Another very remarkable periodical star is that called (3 Persei, 
and also frequently called Algol. It is usually visible as a star of the 
second magnitude, and as such continues for 2 d 13 h .5, when it suddenly 
begins to diminish in splendor, and in about 3 h .5 is reduced to the fourth 
magnitude, at which it continues for about l5 m . It then begins to in- 
crease, and in 3 h .5 is restored to its usual brightness, going through all its 
changes in 2 d 20 h 48 m 58 s .5. Recent observations indicate that this period 
is on the decrease, and not uniformly, but with an accelerated rapidity, 
indicating that it too has its cyclical period, and that instead of continuing 
to decrease, it will after a while be found to increase. 

§ 56*7. The star d Cepheus is also a periodical star. Its period from 
minimum to minimum is 5 d 8 h 47 m 39 s .5. The extent of its variations is 
from the fifth to between the third and fourth magnitudes. Its increase is 
more rapid than its diminution- —the former occupying l d 14 h , and the 
fatter 3 d 19 h . 

§ 568. The periodical star (3 Lyra has a period of 12 d 21 h 53 m 10 8 , 
within which a double maxima and minima take place, the maxima being 
about equal, but the minima not. The maxima are about 3.4, and the 
minima 4.3 and 4.5. Here again the period is subject to change, which 
:« itself periodical. 



152 SPHERICAL ASTRONOMY. 

§ 569. Numerous other periodical stars are recorded. These remark- 
able variations of brightness, and the laws of their periodicity, have sug- 
gested the revolution of some opaque body or bodies around the stars thus 
distinguished, which, becoming interposed at inferior conjunction, would 
intercept a greater or less portion of the light on its way to the earth. Or 
the stars may possess very different' degrees of intrinsic illumination on 
different portions of their surfaces, which, being subject to periodical 
changes and presented to the earth by an axial rotation of the stars, 
would produce the phenomena in question. 

§ 570. Temporary Stars. — The irregularities above referred to may 
afford an explanation of other stellar phenomena, which have hitherto 
been regarded as altogether casual. Stars have appeared from time to 
time in different parts of the heavens blazing forth with extraordinary 
splendor, and after remaining a while, apparently immovable, have faded 
away and disappeared. These are called temporary stars. One of these 
stars is said to have appeared about the year 125 b. c, and with such 
brightness as to be visible in the daytime. Another appeared in a. d. 389, 
near a Aquilse, remaining for three weeks as bright as Venus, and disap- 
pearing entirely. Also in 945, 1264, and 15*72, brilliant stars appeared 
between Cepheus and Cassiopeia, which are supposed to be one and the 
same periodical star, with a period of 312, or perhaps 156 years. The 
appearance in 1572 was very sudden. The star was then as bright as 
Sirius ; it continued to increase till it surpassed Jupiter, and was visible at 
mid-day. It began to diminish in December of the same year, and in 
March, 1574, it had entirely disappeared. So, also, on the 10th of Octo- 
ber, 1604, a star not less brilliant burst forth in the constellation Serpen 
iarhis, which continued visible till October, 1605. 

§ 571. Similar phenomena, though of less splendor, have taken place 
more recently. A star of the fifth magnitude, or 5.4, very conspicuous to 
the naked eye, suddenly appeared in the constellation Ophiuchus. From 
the time it was first seen it continued to diminish, without alteration of 
place, and before the advance of the season put an end to the observations 
upon it, had become almost extinct. Its color was ruddy, which was 
thought to have undergone many remarkable changes. 

§ 572. The alternations of brightness of r\ Argus are very remarkable. 
In 1677 it appeared as a star of the fourth, in 1751 of the second, in 1811 
and 1815 of the fourth, in 1822 and 1826 of the second, in 1827 of the 
first, and in 1837 of the second magnitude. All at once, in 1838, it sud- 
denly increased in lustre so as to surpass all the stars of the first magnitude 
except Sirius, Canopus, and a Centauri. Then it again diminished, but not 



STARS. 



153 



below the first magnitude, till April, 1843, when it had increased so as to 
surpass Canopus, and nearly equal Sirius. 

§ 573. On careful re-examination of the heavens, and comparison of 
catalogues, many stars are missing. 

§ 574. Double Stars. — Many of the stars when examined through the 
telescope appear double, that is, to consist of two individuals close to- 
gether. They are divided into classes according to the proximity of their 
component individuals. The first class comprises those only of which the 
distance does not exceed 1" ; the second those in which it exceeds 1", but 
falls short of 2" ; the third those in which it ranges from 2" to 4" ; the 
fourth from 4" to 8" ; the fifth from 8" to 12"; the sixth from 12" to 
16" ; the seventh from 16" to 24" ; and the eighth from 24" to 32". 

Each of these classes is subdivided into two others, called respectively 
conspicuous and residuary double stars. The first comprehends those in 
which both individuals exceed the 8.25 magnitude, and are therefore sep- 
arately bright enough to be seen with telescopes of very moderate capa- 
city ; the second embraces those which are below this limit of visibility, 
Specimens of each class will be found in the following 

Table. 



y Coronae Bor. 
y Centauri. 
y Lupi. 
e Arietis. 
$ Herculis. 



Class I. 
v Coronas. 
v Herculis. 

A Cassiopeia. 
A Ophiuchi. 
it Lupi. 



)" TO 1". 

7 Ophiuchi. 
<p Draconic. 
(p Ursos Majoris. 
% Aquilae. 
j) Leonis 



Atlas Pleiadum, 
4 Aquarii. 
42 Com®. 
52 Arietis. 
66 Piscinm. 



y Circim. 
i Cygni. 
s Chamasleontis 



Class II.— 1" to 2". 

5 Bootis. | Ursa Majoris. 



i Cassiopeias. 
t 2 Caneri. 



•k Aquilae. 

a Coronas Bor. 



2 Camel opardi. 
32 Orionis. 

52 Orionis. 



Class III.— 2" to 4". 



a Piscium. 
Hydras. 
y Ceti. 
y Leonis. 
y Coronas A us. 



a Crusis. 
a Herculis. 
a Geminorum. 
$ Geminorum. 
5 Coronas Bor. 



v Virginia. 
6 Serpentis. 
e Bootis. 
e Draconis. 
£ Hydras. 

Class 

Phosnieis. 
k Cephei. 
A Orionis. 
H Cygni. 
f Bootis. 



IV.— 



S, Aquarii. 
5 Orionis. 
i Leonis. 
i Trianguli. 
k Leporis. 

■4" to 8". 

£ Cephei. 
it Bootis. 
p Caprieorni. 
v Argus, 
u Aurigas. 



H Draconis. 
fx Canis. 
p Herculis. 
<r Cassiopeia?. 
44 Bootis. 



fi Eridani. 
70 Ophiuchi. 
12 Eridani. 
32 Eridani. 
95 Herculis. 



SPHERICAL ASTRONOMY. 



Orionis. 
y Arietis. 
y Delphini. 



a Centauri. 
6 Cephei. 
B Scorpii. 



a Canum Ven. 
c Normae. 
5 Pi^cium. 



6 Herculis. 
ti Lyras. 

i Cancri. 



Class V.— 8" to 12". 
p Antilae. 
7} Cassiopeiae. 
Eridani. 

Class VL— 12'' to 16" 

v Volantis. 

t) Lupi. 

p Ursae Majoris. 

Class YII.— 16" to 24". 

Serpentis. 
k Coronae Aus. 
j£ Tauri. 

Class VIII.— 24" to 32" 

k Hercnlis. 
k Cephei. 
ip Draconis. 



i Orionis. 
f Eridani. 
2 Canum Ven, 



k Bootis. 
8 Monocerotis. 
61 Cygni. 



24 Comae. 
41 Draconis. 
61 Ophiuchi. 



X Cancri. 
23 Orionis. 



§ 575. Triple, Quadruple, and Multiple Stars. — Stars which answei 
to these designations also occur, and of them the most remarkable are, 



a Andromedse. 


9 Orionis. 


£ Scorpii. 


£ Lyra. 


p Lupi. 


11 Monocerotis. 


£ Cancri. 


ju Bootis. 


12 Lyncis. 



Of these, a Andromedce, \h Bootis, and |x Lupi, appear through telescopes 
of considerable optical power only as ordinary double stars ; and it is only 
when excellent instruments are used that their companions are subdivided 
and found to be extremely close double stars, s Lyra offers the remarka- 
ble example of a double-double star. In telescopes of low power it ap- 
pears as a coarse double star, but on increasing the power, each individual 
is perceived to be double, the one pair being about 2". 5, the other about 
3" apart. Each of the stars £ Cancri, g Scorpii, 11 Monocerotis, and 
12 Lyncis, consists of a principal star closely double and a smaller and 
more distant attendant; while & Orionis, (Fig. 11, of plate,) presents four 
brilliant principal stars of the 4th, 6th, 7th, and 8th magnitudes, forming 
a trapezium, of which the longest diameter is 24". 4, and accompanied by 
two excessively minute and very close companions, to perceive both of 
which is one of the severest tests that can be applied to a telescope. 

§ 5*76. Of the delicate subclass of double stars, or those consisting of 
very large and conspicuous double stars, accompanied by very minute 
companions, the following are specimens, viz. : 



Plate VHL 




TO PH07NTT FAGS 154> 







STARS. 




o 2 Cancri. 


a Polaris. 


k Circini. 


<p Virginis. 


a 2 Capricorni. 


/? Aquarii. 


k Getninorum. 


X Eridani. 


a Indi. 


y Hydrse. 


p Persei. 


16 Aurigse. 


a Lyra. 


i Ursa Major. 


7 Bootis. 


91 Ceti. 



155 



§ 577. Binary Stars. — Many of the double stars are physically con- 
lected in such proximity to one another as to revolve about their common 
centre of gravity in regular orbits. These are called Unary stars. They 
liffer from what are called ordinarily " double stars" in being so near to 
one another as to be kept asunder only by a rotary motion about a com- 
mon centre ; whereas the individuals of a double star are separated by a 
vast distance, and appear double only in consequence of one being almost 
directly behind the other as seen from the earth. 

§ 578. The position micrometer gives from time to time the apparent 
distance between the places into which the stars of a binary system are 
projected upon the celestial sphere, and also the 
angle which the arc of a great circle, drawn from 
one to the other, makes with the meridian passing /\\ /\ 

through either, assumed as the central body ; from Ifee.r^^^y/f'i re . 
these polar co-ordinates, the apparent orbit, as pro- ^-X- ^ 

iected upon the celestial sphere, is easily traced. 

§ 579. The relation which is found to connect the distances with the 
angular velocities shows the stars to be under the control of a central 
force, and the elliptical form of the orbit, with the eccentric position of 
the central star, is proof that this force can be no other than that of grav- 
r tatio'n. 

§ 580. Thus, the same principle which, under the influence of distance, 
directs the satellites about their primaries, and the primaries about our sun, 
also wheels distant suns around suns, each, perhaps, carrying with it its 
system of planets, and each planet a group of satellites. 

§ 581. From the micrometrical measurements above referred to, and 
the intervals of time between them, the elements of the actual stellar 
orbits are easily computed.* A number of sets are given in the following 
table : 



*• See Memoirs of Royal Astronomical Society, vol. v., p. 171. 



156 



SPHERICAL ASTRONOMY. 



o 
o 












1-3 








- 

•"3 






4 


>> 














g 


ft 










"3 




























o 

X 


B 


Si 

I 


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STARS. 157 

§ 582. If the annual parallax of the system, the apparent semi-axis 
of the stellar orbits, and the earth's radius vector, be substituted respec- 
tively for P, s, and p, in Eq. (29), d will become the number of linear 
units in the mean distance between the stars. 

Assuming the data of the table, selecting a Centauri, and making 
a = 15 ".5 and P = 0.913, we have 

whence the stellar orbits of a Centauri are (§ 422) about nine-tenths that 
of Uranus. 

§ 583. Denoting by T the periodic time of a body about its centre, w«j 
have, AnalyLjfechanics, § 201, 

1 ~ k ' 

in which a is the mean distance, if the ratio of the circumference to diam- 
eter, and k the intensity of the central attraction at the unit's distance. 
For a second body 

1 ~ k> ' 
whence 

but from the laws of gravitation k and k' are directly proportional to the 
attracting masses, and we have 

M ' M' : : — — ' 

u.u. . ik . . . ^ f2 . 

Making a = p, a' =. d = 16.977 p ; T~ 1 year, and T'= 77 years ; then 
will M denote the mass of the sun and M' that of the central star of 
a Centauri, and we have from the above proportion 

that is, the mass of the central star is a little over eight-tenths that of 
: ur sun. 

§ 584. Color of Double Stars. — Many of the double stars present the 
curious phenomena of complementary colors. In such instances the larger 
star is usually of a ruddy or orange hue, while the smaller one appears 
blue or green. The double star j Cancri presents the beautiful contrast of 



158 SPHERICAL ASTRONOMY. 

yellow and blue ; y Andromeda?, crimson and green. Where there is 
great difference in the magnitudes of the individuals, the larger is usually 
white, while the smaller may be colored ; thus, i\ Cassiopeia exhibits the 
beautiful combination of a large white star and a small one of a rich ruddy 
purple. If this be not the mere optical effect of contrast of brightness, 
what variety of illumination two suns — a red and a blue one, a crimson 
and green one — must afTord to the inhabitant* of planets that circulate 
around them, having sometimes both suns above their horizon at once and 
at -others each in succession, thus producing an alternation of red and blue, 
crimson and green days! Insulated stars of a reC color, almost as deep as 
blood, occur in many parts of the heavens. 

§ 585. Proper Motions of the Stars. — As might be expected from their 
mutual attractions, however enfeebled by distance and opposing attractions 
from opposite quarters, the stars are found to have a proper motion, which 
in the lapse of time has produced a sensible change of internal arrange- 
ment. Thus, from the time of Hipparchus, 130 years b. c, to a. d. 171 7, 
eighteen hundred and forty seven years, the conspicuous stars Sinus, Arc- 
turus, and Aldebaran, are found to have changed their latitudes respect- 
ively 37', 42', and 33', in a southerly direction. Besides, the observations 
of modern astronomy prove that such motions do really exist. The two 
stars 61 Cygni are found to have retained sensibly unchanged their dis- 
tance apart for the last fifty years, while they have shifted their places in 
the heavens in the same interval no less than 4' 23", giving an annual 
proper motion to each of 5 ".3. Of the stars not double, and no way dif- 
fering from the rest in any other sensible particular, s Indi and ,a Cassio- 
peia; have the greatest proper motions, amounting annually to 7 ".74 and 
3 ".74 respectively. 

§ 586. Proper Motion of the Sun. — The inevitable consequence of a 
p'oper motion in our sun, if not equally participated in by the rest, must 
be a slow average apparent tendency of all the stars to the point of the 
celestial sphere from which the sun is moving, and a corresponding retro- 
cession from the opposite point — and this, however greatly individual star? 
may differ from such average by reason of their own peculiar proper mo- 
nan. This is the necessary effect of parallax, and has been detected by 
observation. 

By properly treating the observations on the stars of the northern hemi- 
sphere, the solar apex, as it is called, or the point towards which the sun 
was moving at the epoch of 1790, was in right ascension 250° 09', and 
north polar distance 55° 23'. The southern stars gave, by a similar mode 
of treatment, right ascension 260° 01', and nTth polar distance 55° 37': 







TO FMOJfT VA&hl 159. 



NEBULAE. 159 

results so nearly identical as to remove all doubt of the sun's proper 
motion. 

§ 587. All analogy would 1© to^he conclusion that the sun is de- 
scribing an orbit of vast extent about the%^ntre of gravity of the group of 
stars of which it forms a single member, and of which the milky way is 
to us but the distant trace, while this group ^ay itself be moving as a 
single systen!teu;Wmd .some other and vastly distant centre. A line drawn 
tangent to the solar orbit in 1790 pierced the celestial sphere near the 
stars ir Herculis and a Columba, the sun being tfien moving* towards the 
former and from the latter. And the result of calculations thus far gives 
to the sun a velocity of 422,000 miles a day, or little more than one-fourth 
the earth's rate of annual motion in its orbit. \ 

\ 



NEBULA. 



,r M shimi 



§ 588. Besides the stars which appear a^ shining points, there are 
cloud-like patches of light to be seen scattered here and there over the 
eelestial vault. These are called nebulm. They present themselves under 
great variety of shapes and sizes, as exemplified in Figs. 12, 13, 14 (front- 
ing plate), and exhibit in the telescope different characters of internal 
structure with every increase of optical power. They are very unequally 
distributed over the heavens. In the northern hemisphere, the hours 3, 4, 
5, 16, 17, and 18 of right ascension are singularly poor, while the hours 10, 
1 1, and 12, especially the latter, are exceedingly rich in these objects. In the 
southern hemisphere a much greater uniformity prevails, with two remark- 
able exceptions, to be noticed presently. They have no decided tendency 
to any particular region. 

§ 589. When viewed through the telescope, many nebulae are resolved 
into stars, and the number that thus yield their cloud-like aspect increases 
with every augmentation of instrumental power. Nebulas are therefore 
classified, with reference to their appearance through the telescope, into 
resolvable, irresolvable, planetary, and stellar nebula?, and nebular stars. 

§ 590. Resolvable Nebula?. — These are usually called clusters af stars. 
Some are very broken in outline, while others are so regular as to suggest 
the prevalence of some internal action productive of symmetrical arrange- 
ment among their internal parts. 

§ 591. Irregular clusters are much less rich in stars, and much less 
condensed towards the centre. In some the stars are nearly of the same 
size, in others very different. The group called the Pleiades, in which six 



100 SPHERICAL ASTRONOMY. 

or seven stars may be counted with the naked eye, and fifty or sixty with 
the telescope, is one of the most obvious examples of this class. Coma 
Berenices, represented in Fig. 15, Plate X, is another such group. 

§ 592. Globular Clusters. — -These take their name from their round 
appearance. They are much more difficult of resolution, and some have 
frequently been mistaken tor comets without tails. When viewed through 
the telescope, they are found to be composed of stars so crowded together 
as to occupy an almost? definite outline, and to run up to a blaze of light 
towards the centre, where their condensation is greatest. It would be vain 
to attempt to count the stars in these clusters; some have been estimated 
to contain five thousand, within an area not greater than the tenth part of 
the lunar disk. 

§ 593. Elliptic Nebulae. — The figure here again suggests the name. 
They are of all degrees of eccentricity, from moderately oval to elongations 
so great as to be almost lmear. In all, the density increases towards the 
centre, and generally theilfnlternal strata approach more nearly the spheri- 
cal form than their external. Their resolvability is greater in the central 
parts ; in some the condensation is slight and gradual, in others great and 
sudden. 

The largest and finest specimen of elliptic nebulae is in the Girdle of 
Andromeda, given m Fig. 12, Plate IX. 

§ 594. Annular nebulai also exist, but are very rare. The most con- 
spicuous of this class is found between (3 and y Lyrae, and may be seen 
through a telescope of moderate power. The central vacuity, Fig. 16, 
Plate IV, is not quite dark, but appears as a light-colored gauze stretched 
over a hoop. The powerful telescope of Lord Rosse resolves this nebula 
into excessively minute stars, and shows filaments of stars hanging to 
its edge. 

§ 595. Spiral Nebulai. — These are most curious objects. Their dis- 
covery is but very recent, and is due to the powerful instrument of Lord 
Rosse. As their name indicates, they appear to consist of a spiral or vor- 
ticose arrangement of stars diverging from a centre, and suggest the idea 
of a vast self-luminous mass of matter, travelling to a common destination 
along separate curvilinear paths. Their form and general appearance are 
represented in Figs. 17 and 18, Plate X. 

§ 596. Planetary Nebulae. — These take their name from the planet- 
iike disk which they present. In some instances they bear a perfect re- 
semblance to a planet in this respect, being round or slightly oval, and 
quite sharply terminated. In some the illumination is perfectly equable : 
in others mottled, and of a peculiar texture, as if curdled. They are com- 




TO FRONT -PA&E160. 



Plate XT. 





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FIG 23. 


FIG. 24. 


ia=27l°45'D.=19°56 S 


^=276°15 DfG4 t> 53. 







TO &ROJBTT PA.GE 161. 



NEBULAE 161 

paratively rare, not above four or five and twenty having been observed, 
and of these nearly three-fourths are in the southern hemisphere. They 
are sometimes colored; one, whose right ascension in 1830 was 10 h 16 m 
36 s , and north polar distance 107° 47', is of a sky-blue; it is slightly el- 
liptical, and has an apparent diameter of 30". One of the largest of these 
objects is near fi Ursa Majoris; its apparent diameter is 2' 40", which, 
supposing its distance not greater than 61 Cygni, would imply a linear 
one seven times greater than Neptune's orbit. The light of this stupen- 
dous globe is perfectly equable, except at the edge, where it is slightly 
softened. The Rosse telescope is rapidly transferring the planetary to the 
annular and spiral nebula. 

§ 597. Double Nebula. — These occasionally occur, and the constituents 
are most commonly spherical, and belong, most probably, to the globular 
clusters. Figs. 19 and 20, Plate XI. 

§ 598. Nebulous Stars. — These consist of stars surrounded concentric- 
ally by a perfectly circular disk or atmosphere of faint light, Figs. 21 and 
22, Plate XI; in some cases diminished gradually on all sides, and in others 
suddenly terminated. In right ascensions 7 h 19 m 8 s , 3 h 58 m 36 s , and north 
polar distances 68° 45', 59° 40', are stars of the eighth magnitude sur- 
rounded by photospheres of the kind just described, respectively 12" and 
25" in diameter. 

§ 599. Nebulous Double Stars. — In Fig. 23, Plate XI, is represented an 
elliptical nebula having its longer axis about 50", in which, symmetrically 
placed and rather nearer the vertices than the foci, are the equal individuals 
of a double star, each of the 10th magnitude. In a similar connection, 
Fig. 24, Plate XI, represents two unequal stars situated at the extremities of 
the major axis. In right ascension 13 h 47 m , north polar distance 129° 9', 
is an oval nebula of 2' diameter, having near its centre a close double star 
of the 9.10 magnitude, not more than 2" asunder. Other instances might 
be adduced of nebuhe uniting great peculiarities of shape with regularity 
of outline. 

§ 600. Irregular Nebulw. — Besides the nebula? just described, there is 
a class totally different in character, being of great extent, utterly devoid 
of all symmetry of form, and most remarkable for the extent of their con- 
volutions and the distribution of their light. No two of them can be said 
to present any similarity of aspect, and the only thing in common is 
their locality, which is in or near the borders of the milky way, the most 
remote being that in the sword handle of Orion, about 20° from the gal- 
lactic circle, and represented in Fig. 13, Plate IX But even this is in the 
prolongation of a faint offset of the milky way. 

11 



102 SPHERICAL ASTRONOMY. 

These nebulae may be grouped into four great masses, which occupy the 
regions of Orion, of Argo, of Sagittarius, and of Cygnus. 

§ 601. The Magellanic Clouds, or the Nubecula (Major and Minor), 
as they are called in celestial maps and charts, are two nebulous or cloudy 
masses of light conspicuously visible to the naked eye in the southern 
hemisphere, and in appearance and brightness resemble portions of the 
milky way of the same size. They are in shape somewhat oval, the larger 
deviating most from the circular form. The larger is situated between the 
hour circles 4 h 40 m and 6 h 40 m , the parallel* 156° and 162° north polar 
distance, and occupies an area of about 42 square degrees. The lesser, 
which is between the hour circles h 28 m and l h 15 m , and the parallels of 
162° and 165° north polar distance, covers about ten square degrees. The 
general ground of both consists of large tracts of nebulosity in every stage 
of insolubility, from light irresolvable up to perfectly separated stars like 
the milky way, including groups sufficiently insulated and condensed tc 
come under the designation of irregular and globular clusters, the latter 
beiug in every stage of condensation. In addition they contain nebular 
objects quite peculiar, and which have no analogy in any other part of the 
heavens. Globular clusters, except in one region of small extent, and neb- 
ulae of regular elliptic forms are comparatively rare in the milky way, but 
are congregated in greatest abundance in parts of the heavens the most re- 
mote possible from the gallactic circle ; whereas in the Magellanic Clouds 
they are indiscriminately mixed with the general starry ground. 

§ 602. Regarding the nubecula as spherical in form, and not as vastly 
long vistas foreshortened by having their ends turned towards the earth — 
which would be improbable seeing there are two of them close together — 
the brightness of objects in their nearer portions cannot be much exagger- 
ated, nor those in its remoter much enfeebled by difference of distance. 
It must, therefore, be an admitted fact that stars of the 7th and 8th mag- 
nitudes and irresolvable nebulae may coexist within limits of distance com- 
paratively small, and that all inferences in regard to relative distance 
drawn from relative magnitudes must be received with caution. 

§ 603. Our Sun a Nebulous Star. — Various phenomena indicate that 
our sun is itself a nebulous star. The chief is that called the zodiacal 
light, which may be seen on any clear evening soon after sunset about the 
months of March, April, and May, and at the opposite seasons of the year 
just before sunrise, as a conically-shaped light, extending from the hori- 
zon upwards in the direction of the sun's equator. The apparent angular 
distance of its vertex Ffrom the sun S varies from 40° to 90°, and its 
breadth at its base, perpendicularly to its ler.gth, frcm 8° to 30°. Every 



~x 








TOFRCWT PAGE 163 



NEBTJLJE. 



163 



circumstance connected with it indicates it to be Fis - 9a 

a lenticularly-formed envelope^ surrounding the f^ 

sun, and extending beyond the orbits of Mercury Hori zon 




and Venus and even to the Earth, 
having been seen 90° from the sun 
circle. Different parts of the heavens 
examples of similar forms. 'Figs. 25, 26, 27,', 
riateX*Ov -. \ 

§ 604. Tk&mHes. — Nothing prevents that the Articles of this vast ma- 
terial envelope\nay have tangible size and be at £tfeat distances apart, and 
yet compared wri^i the planets, so called, be bur as dust floating in the 
established fact that masses of stone and lumps of 



sunbeam. It is 

iron, called Aeroldte\ do occasionally fall upon the earth f.om the upper 
-regions of thevatmJsphere, and that they have done so since the earliest 
records. On the-2cth j^pril, 1803, one of these bodies foil in the imme- 
diate vici^y of*tKe town of L'Aigle, in Normandy, and by its explosion 
into fragments, scattered thousands of stones over an area of thirty square 
miles. Four instances are recorde4 of persons having been killed by the 
descent of such bodies, and after -every vain attempt to account for them 
as coming originally from ihA earth,* and even from the moon, by volcanic 
projections, their planetary nature is now generally admitted. Their heat 
when fallen, the igneous phenomena which accompany them, their explo- 
sion on reaching the denser regions of our atmosphere, are accounted for 
by the condensation in front of them created by their enormous velocity, 
and by the relations of air, in a highly attenuated state, to heat. 

§ 605. Meteors, — Besides these more solid bodies, otheis of much less 
density appear also to be circulating around the sun at the distance of the 
earth from that luminary. These on coming within the atmosphere ap- 
pear as shooting stars, followed by trains of light, and are called Meteors. 
They appear now and then as great fiery balls, traversing the upper le- 
gions of the atmosphere, sometimes leaving long luminous trains behind 
them, sometimes bursting with a loud explosion, and sometimes becoming 
quietly extinct Among these latter may be mentioned the remarkable 
meteor of August 18th, 1783, which traversed the whole of Europe, from 
Shetland to Rome, with a velocity of 30 miles a second, at a height of 50 
miles above the earth, with a light greatly surpassing that of a full moon, 
and diameter quite half a mile. It changed its form visibly and quietly, 
separated into several distinct p#rts, which proceeded in parallel direc- 
tions, each followed by a train. 

§ 606. On several occasions meteors have appeared in astonishing 



1^4 SPHERICAL ASTRONOMY. 

numbers, falling like a shower of rockets or flakes of sikw, 
at once whole continents and oceans, even in both hemispheres. And it 
is significant that these displays have occurred between the 12th and 14th 
November and 9th and 11th August. In November they are much more 
brilliant, but their returns /less certain than in August, when numerous 
large and brilliant shooting-stars with trains are almost sure to be seen. 

§ 607. Annual periodicity, irrespective of geographical location, points 
at once to the place of the earth in its orbit as a necessary concomitant, 
and leads to the conclusion that at that place the earth enters a stratum, 
or annular stream of meteoric planets, in their progress of circulation 
around the sun. The earth plunging in its annual course into a ring of 
these bodies, and of such thickness as to be traversed m a day or two, their 
motions, referred to the earth as at rest, would be sensibly uniform, recti- 
linear, and parallel. Viewed from the centre of the earth, or from any 
point on its surface, neglecting the diurnal as b,eing insignificant in com- 
parison with the annual motion, their paths wojild appear to ^erge from 
a common point on the celestial sphere. Now this is precisely what hap- 
pens. The vast majority of the November meteors appear to describe arcs 
of great circles passing through y Leonis, and those of August appear 
to move along paths having a common point in (3 Camelopardi. 

§ 608. As the ring may have any position and be of an elliptical fig- 
ure having any reasonable eccentricity, both the velocity and direction of 
each meteor may differ to any extent from those of the earth, so there is 
nothing in the great difference of latitude of these meteoric apices at all 
opposed to the foregoing conclusion. 

§ 609. If the meteoric planets were uniformly distributed in the sup- 
posed ring, the earth's annual encounter with them would be certain if it 
occurred once; but if such ring be broken, and the bodies revolve in 
groups, with periods differing from that of the earth, years may pass with- 
out rencontre, and when such happen, they may differ to any extent in 
intensity of character, according as the groups encountered are richer or 
poorer in the number of their elements. 

§ 610. From careful observations, made at the extremities of a base 
50,000 feet long, it has been inferred that the heights of meteors at the 
instant of first appearance and disappearance, vary from 16 to 140 miles, 
and their relative velocities from 18 to 36 miles a second. Altitudes 
and velocities so great as these clearly indicate an independent planetary 
circulation round the sun. 

§ 611. It is not impossible that some of these bodies may have been 
converted by the superior attraction of the earth, arising from greater prox- 



0, /^.^/^A^TW^ 



EPHEMERIDES. 165 

unity, into permanent satellites ; and there are those who believe in the 
existence of at least one of these bodies, which completes its circuit about 
the earth in about 3 h 20 m , and therefore at a mean distance of about 
5000 miles. 

EPHEMERIDES. 

§ 612. The facts and principles now explained enable us to predict the 
aspect of the heavens, or positions of the heavenly bodies, for all future 
time. This prediction is usually drawn up in the condensed form of tables, 
which are called epkemerides. The table relating to any one body is called 
the epheweris of that body, as the ephemeris of the sun, of the moon, &c. 

§ 613. Ephemerides are prepared in advance to subserve the wants and 
promote the interests of navigation, geography, and chronology, as well as 
of future astronomical discovery and research. 

§ 614. To facilitate the computation of the ephemeris of a body, it ia 
usual first to construct what are called its tables ; and the manner of doing 
this may best be explained by taking a particular example, say that of the 
sun, or rather the earth, since this is the moving body ; but as the place of 
the sun, as seen from the earth, differs from that of the earth as seen from 
the sun by the constant 180°, we shall speak of the sun. 

§ 615. We have seen, § 197, how the mean longitude of the sun, his 
mean motion, longitude of the perigee, and eccentricity, may be found 
from observation and computation. These elements being found at epochs 
widely separated from one another, the changes which take place in the 
last three, and the rate of motion of the perigee, are ascertained. 

§ 616. Having fixed upon any epoch, say mean noon or midnight, 1st 
January, 1800, any interval of time, either after or before the epoch, mul- 
tiplied by the mean motion of the sun in longitude, will give the increase 
of mean longitude during that interval, and being added to the mean lon- 
gitude at the epoch and the sum divided by 360°, the remainder will give 
the mean longitude at the beginning of the interval, if it be before, or end, 
if it be after the epoch. These longitudes, with the corresponding dates, 
being tabulated, give what is called a table of epochs, which tells by simple 
inspection the mean longitude on any given day, hour, minute, and second. 

§ 617. The same process being performed with reference to the longi- 
tude of the perigee and its rate of change, gives a corresponding table in 
which the longitude of the perigee is found. 

§ 618. Resuming Eq. (o), Appendix No. V., and causing mt!, which 
is the mean anomaly, to vary from 0° to 360°, corresponding" equations of 



1QQ SPiitKICiL ASTRONOMY. 

the centre will result, and these properly arranged form a table of equations 
of the centre, of which the arguments, as they are called, are the mean 
anomalies. Then causing the eccentricity to vary according to ascertained 
rates, the same equation gives the elements of an additional table by which 
the equations of the centre may be corrected from time to time. 

§ 019. Nutation causes the true equinox to oscillate about a mean 
place, its distance therefrom being equal to the algebraic sum of two func- 
tions, of which one depends upon the longitude of the moon's node, the 
other upon the longitude of the sun, and both upon the obliquity of the 
ecliptic. Tables containing the values of these functions for assumed places 
of the moon's node and of the sun, give the numbers whose sum is equal 
to the equation of the equinoxes in longitude. 

§ 620. In addition, the larger of the planets, especially Venus and Ju- 
piter, disturb the earth's orbit. These perturbations are computed by pro- 
cesses in physical astronomy, and their values arranged under heads that 
give the angular distances of the disturbing planets from the earth as seen 
from the sun, and, together with the place of the moon's node, furnish the 
arguments with which other tables are entered that give the corresponding 
effects upon the sun's longitude. 

§ 621. Lastly, as the purpose is to find the place where the sun's centre 
is to be seen, provision is made for the effect of aberration. This in 
the case of the sun is nearly constant, and equal to — 20".25, because 
of the small eccentricity of the earth's orbit, the greatest variation there- 
from being less then 0".35. This constant is included in the epoch tables. 

§ 622. JEphemeris of the Sun. — We are now prepared to find where the 
sun has been and where he will be on the celestial sphere throughout time. 
For this purpose, enter the table of epochs with the date, take out his mean 
longitude and the longitude of the perigee ; the difference will be the mean 
anomaly, with which enter the table of the equations of the centre and 
take out the corresponding equation ; add this to or subtract it from the 
mean longitude according to its sign, and the result will be the true lon- 
gitude of the sun as affected by nutation and perturbations. Take these 
latter from the appropriate tables, and we have 

True longitude of sun = mean longitude + equation of the centre + 
nutation or equation of equinoxes in longitude -{-perturbations. 

§ 623. With the true longitude and obliquity of the ecliptic, we pass, 
by spherical trigonometry, § 149, to right ascension and declination. 

§ 624. The mean anomaly in Eq. (?i), Appendix V., gives the corres- 
ponding true anomaly ; and the latter in Eq. (c), same Appendix, gives the 



EPHEMERIDES. 



167 



radius vector r, which in equations (28) and (29) give the correspond- 
ing horizontal parallax and apparent diameter. 

§ 625. The mean longitude corrected for the equation of the equi- 
noxes in right ascension, and diminished by the right ascension, gives 
the equation of time. 

§ 626. These and other elements being determined at different epochs, 
say for every noon on some fixed meridian, their consecutive differences, 
divided by the number of hours between the epochs, give the hourly 
changes, and therefore the means of finding the value of the elements 
themselves for any other meiidian. 

The elements with their hourly changes make up, when properly tabu- 
lated, an ephemeris of the sun. 

§ 627. Ephemeris of the Moon. — The motion of the moon is altogether 
moi'e irregular and complicated than the apparent motion of the sun, owing 
mainly to the disturbing action of this latter body. But these and other 
perturbations have been computed and tabulated, and from these tables, 
including those of the node and inclination, the places of the moon in her 
orbit are found in much the same way as those of the sun in the ecliptic. 
The mean orbit lonc/itude of the moon and of her perigee are first found 
and corrected : their difference gives her mean anomaly, opposite to which 
in the appropriate table is found the equation of the centre, and this being 
applied with its proper sign to the mean orbit longitude gives the true 
orbit longitude. 

§ 628. Let E be the earth, if the moon, 
V the vernal equinox, VM' an arc of the 
ecliptic, VQ of the equinoctial, and MM'of a 
circle of latitude; then will MM' be the 
latitude and VM' the longitude of the moon, 
VJV the longitude of the node and VEN 
-j- N EM the orbit longitude of the moon. 

Subtracting from the orbit longitude of 
the moon the longitude of the node, the re- 
mainder NM will be the moon's angular distance from her node. This 
and the inclination M N M' will give, in the right-angled triangle MNM, 
the latitude MM and the side N M' , which latter added to the longitude 
of the node N V gives the longitude VM'. The latitude* and longitude, 
together with the obliquity of the ecliptic, give, § 153, the right ascension 
and declination. The radius vector, equatorial horizontal parallax, apparent 
diameter, &c, are computed as in the case of the sun. And thus an 
ephemeris of the moon is constructed. 



Fig. 99. 




168 



SPHERICAL ASTRONOMY 



§ 6*29. Ephemeris of a Planet. — From tables of a planet its true orbit 
longitude as seen from the sun is found, as in the case of the moon as seen 
from the earth. From the heliocentric orbit longitude, heliocentric longi- 
tude of the node, and inclination, the heliocentric longitude and latitude, 
together with the radius vector, are found ; just as the corresponding geo- 
centric elements of the moon are found from similar data relating to the 

lunar orbit ; and from the heliocen- 

• , • i i • , i Fig. too. 

trie longitude, latitude, and radius 

vector, we pass to the geocentric, 

thus: 

§ 630. Let P be the planet, E 

the earth, S the sun, and the 

projection of the planet upon the 

plane of the ecliptic. Draw from 

S and E the parallels S V and 

E V to the vernal equinox, and 

make 

r = E S = radius vector of earth ; 

r' = SP — radius vector of planet; 

X = VS == heliocentric longitude of planet ; 

X' = V E = geocentric longitude of planet; 
6 = P S = heliocentric latitude of planet ; 

6' = PEO = geocentric latitude of planet; 

S = OS E = commutation ; 

= S E = heliocentric parallax ; 

E = SEO = elongation ; 

O = V'ES = longitude of sun. 




Then 

and because 

we have 



S = r' cos & ; 

VST= V'ES =360°- 0, 

S= 180°— (360°- 0) — \= - 180° -X; 
whence the commutation is known. Then in the plane triangle OES, 
r' cos d + r : r' cos 6 — r : : tan \ (E + O) : tan \ (E — O) ; 

S+ O + E=180°, 



but 
whence 



\(E+0) = 90°- | 



(163) 



EPHEMEEIDES. 1G3 

Substituting this above, we have 

, / -n ^x , « r ' cos ^ — r 

tan i (^- 0) = cot i £ . - -■— ; 

2 v ' 2 r' cos d + r 



and making 

r' cos 



tan x 



tan 1 (E- 0) = cot 1 £ . ta " X -^ = cot4£.tan( x - 45 c ) . . (164) 

Knowing from Eq. (163) the half sum of E and 0, and from Eq. (164) 
their half difference, E and become known. 
And we have 

\' = E- (360°- ©) = ^+ O - 360° . . . (165) 

§ 631. Again; 

P = E . tan &' = S . tan 6 ; 
whence 

tan &' SO sin E 



tan 6 EO sin S ' 

and 

. sin ^ 
tan 6' = tan 6 . - — a ...... (166) 

sin S x 

From equations (165) and (166) the geocentric longitude and latitude be- 
come known. 

§ 632. Denote by r" the distance E P of the planet from the earth ; 
then will 

E —. r" cos ^ and SO — r'cos 6 ; 

and in the triangle E S 

r" cos 6' : r'cos d : : sin S : sin E\ 

whence 

. cos sin S . 

r" = r'. -.- — - (167) 

cos &' sin E v ' 

The right ascension, declination, horizontal parallax, and apparent diam- 
eter, are found as in the case of the sun and moon. 

§ 633. The ephemerides most commonly used in this country are those 
computed for the meridian of Greenwich, England, and published several 
years in advance under the title, " Nautical Almanac and Astronom- 
ical Epiiemeris." 



170 SPHERICAL ASTRONOMY. 



CATALOGUE OF STARS. 



§ 634. Another important, indeed indispensable auxiliary to practka. 
astronomy, is a catalogue of stars. This consists of a list of certain stars 
arranged in the order of their right ascensions, with the means of obtaining 
the right ascensions and declinations of the places in which they appear at 
any given epoch. T2* 

§ 635. By precessiou, § 157, nutation, § 156, and aberration, § 215. 
the right ascension and declination of a star are ever varying. 

The place of a star referred to the mean equinoctial and mean equinox 
is called its mean place; that, referred to the true equinoctial and true 
equinox, its true place ; and that in which it is seen referred to the true 
equinoctial and true equinox, its apparent place. 

The true place is equal to the mean, corrected for nutation ; and the 
apparent place is equal to the true, corrected for aberration. The true and 
mean places are found from the apparent, by applying the same correc- 
tions, with their signs changed. 

§ 636. The apparent places of the stars are used as points of reference 
on the celestial sphere ; and knowing the right ascensions and declinations 
of these places, those of the apparent place of any other object become 
known also when the distance of the latter in right ascension and declina- 
tion from one or more stars is found by instrumental measurement. 

§ 637. Annual Precession. — The annual precession for any year is, 

Luni-solar = 50".37572 — y x 0".00024358£O, ' 
General = 50",21129 + y x 0".0002442966 ; 

in which y denotes the number o£ the year from 1750, minus when before 
that epoch. 

§ 633. The epoch of the catalogue Vhich will be referred to hereafter, 
that of the British Association, is Januaryslst, 1850. Making y — 100, 
and denoting the nutation of obliquity by 4 w,"we have 

A w = 9".2500 cos Q - 0\0903 cos 2 Q -f- 0".0900 cos 2 ]) -f- 0^.5447 cos 2 © ; 

in which Q denotes the mean longitude of the moon's node, X> the true 
longitude of the moon, and the longitude of the sun. 

§ 639. And assuming the mean obliquity of the ecliptic for 1850 equal 
to w = 23° 27' 31", we have then for the nutation in longitude, denoted 
by J L, 

a£=- 17".3017 sin Q -f 0".2081 sin 2 Q - 0".2074 sin 2 J) - 1".2552 sin 2 



CATALOGUE OF STARS. -tf\ 

§ 640. Denoting the equation of the equinoxes in right ascension by 
d A, we have 
A A = — 15".872 sin ft + 0".192 s.n 2 ft — 0".190 sin 2 D - l".50O sin 2 ©. 

'§ 641. Denoting the right ascension and declination of any body by 
a and 5 respectively, and by p and p', its change in the same due to an- 
nual precession, then will 

p = 46".05910 -f£J^Q5472 sin a . tan 5 . . . (168) 

y = 20".05472 cos a (169) 

§ 642. The change in right ascension and declination for any fractional 
portion of the year will be found by multiplying the above by 

t= = 0.00273785 x d . . . . (110) 

365.25 k 7 

In which d denotes the number of days from the beginning of the\year to 
the end of the fraction. S^^^S, 

§ 643. Denoting by 4 a, and 4 5, the change in right ascension and dec- 
lination arising from nutation, then, omitting terms involving sin 2 Di, will 

A a t = — (15".872+ 6".88S sin a. tan i). sin ft — 9". 250 cos a . tan S . cos ft ) 

-f- (o".191+0".083 sin a . tan i) . sin 2ft-f0".090 cos a. tan i. cos 2^ [■ (171) 
— (1".151+0".500. sin a. tan <5). sin 2© — 0". 545 cos o. tan 5. cos 2© ) 

A 5, = 9". 250 . sin a . cos ft — 6". 888 cos a . sin & J 

— 0".090sinacos2 ft + 0".083 cos a .sin 2 ft >■ .(172) 
+ 0".545 sin a . cos 2 © — 0".500 cos a . sin 2 O ) 

§ 644. Aberration. — Denoting by 4 a 2 and 4 5 2 the change in right 
ascension and declination arising from aberration, disregarding the eccen- 
tricity of the earth's orbit, 

Aa 2 ~ — (20".4200 sin © . sin a -f 18".7322scos © cosa).sec<J . . (173) \ 
Ah= — (20''.4200 sin © . cos a — 18".7322 cos © sin a) sin 



'} 



(174) 
8'. 1289 cos © cos S 

§ 645. Multiplying Eq. (168) by Eq. (170), adding together the prod- 
uct and equations (171) and (173), and denoting the apparent right as- 
cension by a and the mean by a', there will result, after suitable reduction, 

a' — a = Aa = (t — 0.843 sin fo + 0.004 sin 2 Q — 0.025 sin 2 © ) X (46".059 + 20".066 sin a tan S) 

- (9".250 cos ft - 0''.090 cos 2 ft + 0".545 cos 2 © ) . cos a . tan i 

— 20".42O sin © . sin a . sec 6 

— 18".732 cos © . cos o . see <5 

- ■ 0".0530 sin ft -f 0".000 sin 2 ft — 0'.0039 sin 2 ©. 



172 SPHERICAL ASTRONOMY. 

Multiplying Eq. (169) by Eq. (170), adding together the product and 
equations (172) and (174), and denoting the apparent declination by 8 and 
the mean by 8', we also have, after reduction, 

$: - 6 — a & = (t - 0.343 sin Q + 0.004 sin 2 Q - 0.025 sin 2 © ) X 20''.055 cos a 
+ (9".250 cos Q, - 0".090 cos 2 Q -f 0".545 cos 2 ©) sin a 

— 20".420 sin © . cos a . tan 6 

— 18".732 cos (tan w . cos h — sin a . sin <5). 

Neglecting the three last terms in the value for A a as insignificant, and 
making 

A— — 18".732 cos 0, 

B= — 20".420 sin 0, 

C—t— 0.025 sin 2 — 0.343 sin Q, + 0.004 sin 2 ft, . 

D= — 0".545 cos 2 O — 9".250 cos Q + 0".090 cos 2 ft, 

a = cos a . sec #, 

6 = sin a . sec #, 

c = 46".059 + 20".055 sin a . tan 8, 

d = cos a tan 5, 

a' = tan w . cos 8 — sin a . sin 8, 
b' = cos a . tan £, 
c' = 2 0".0 5 5 cos a, 
cT = — sin a ; 

the above become 

AoL = a.A + b.B + c.C+d.D .... (175) 
A8 = a'.A + b , .B + c'.C-\-d'.D .... (176) 

§ 646. Proper Motion. — To the foregoing must be added the proper 
motion of the star when it is known with sufficient accuracy, and is of 
sufficient magnitude to be taken into the account. 

Equations (173) and (174) enable us to pass from the apparent to the 
true, or from the true to the apparent right ascension and declination of a 
star. 

§ 647. Since the motion of the equinoxes is very slow, the values of the 
functions a, 6, c, d, a', b', c', and d / will be sensibly constant for a number 
of years, particularly when the stars are not very near the poles, while 
those of the functions A, B, C, and D vary sensibly from day to day 
These latter are, therefore, computed for every day in the year, and their 
logarithms recorded in the astronomical ephemeris ; the others are com- 
puted for the epoch of the catalogue, and their logarithms recorded oppo- 
site each star in the catalogue. 



CATALOGUE 0/ STARS. 173 

§ 648. Construction of the Catalogue. — The elements relating to each 
star occupy a portion of the two pages exposed to view on opening the 
catalogue On the left-hand page will be found every thing relating to 
right ascension, and on the right, to declination. The left-hand page con- 
sists of eleven vertical columns : in the first is placed the number of the 
star, in the order of its right ascension ; in the second, the name of the con- 
stellation in which it is situated, with its letter or number; in the third, 
its magnitude; in the fourth, its mean right ascension, January 1st, 1850, 
in time ; in the fifth, its mean annual precession in right ascension, Eq. 
(168), reduced to time ; in the sixth, its secular variation, reduced to time ; 
in the seventh, its proper motion in right ascension, reduced to time ; and 
in the eighth, ninth, tenth, and eleventh, the logarithms of the functions 
«, 6, c, and d, reduced to time, respectively, each preceded by the sign of 
the function to which it belongs. The right-hand page consists of fifteen 
vertical columns, in the first of which the number of the star is repeated ; 
the second contains the mean north polar distance, January 1st, 1850; 
the third, fourth, and fifth, the annual precession, secular variation, and 
proper motion in north polar distance, respectively ; the sixth, seventh, 
eighth, and ninth, the logarithms of the functions a\ b r , c\ and d' re- 
spectively, each preceded by the sign of the function to which it be- 
longs ; the remaining columns contain the numbers by which the star is 
recognized in the catalogues of the several authors, whose names are at 
the top. 

Example. — Required the apparent right ascension and declination oi 
y Orionis, February 5th, 1854. 

h. m. s. o ' 

Mean a January 1st, 1850 . 5 17 05.33 Mean K P. D. . . . 83 47 25.7 

4 years' prec. and pr. motion -f* 12.88 4 y'rs' prec. and pr. motion — 14.9 

Meana January 1st, 1854 . 5 17 18.21 Mean N. P. D 83 47 10.8 



a 
A 

a A 

b 
B 

bB 



Logs. Nat. Nos. Logs. Nat. Nos. 

+ 8.0963 a' . — 9.5120 

- 1.1363 A . - 1.1363 



-9.2326 . — OUT! a' A . +0.6483 . + 4".449 



+ 8.8188 V . — 8.3039 

+ 1.1443 B . + 1.1443 



+ 9.9631 + 0.919 VB . - 9.4482 . - 0.281 



174 



SPHERICAL ASTRONOMY. 



c 
C 

cC 

d 
D 



4- 0.5070 
— 9.2812 



— 9.7: 



-f 7.1304 
— 0.5713 



7.7oi7 



Nat Nos. 



- 0.614 



0.005 



Act = -f- 0.129 



c' 
C 

<?C 

d' 
D 

d'D 



Logs. 

— 0.5721 

— 9.2812 

+ 9.8533 

+ 9.9923 

— 0.5713 

— 0.5836 



Nat Noa 



+ 0.713 



— 3.661 



AN.P.D. = + 1.220 



Hence app't right aseeusicn, Feb. 5, 1854, 5 h I7 m 18 s .21 + 8 .13 = 5 h I7 m 18 s .34 
app't N. P. D 83° 47' lu".80-f 1".22 = 83° 47' 12".02 



APPLICATIONS. 



TIME OF CONJUNCTION AND OF OPPOSITION. 

§ 649. To find from the ephemeris the time at which two bodies are in 
conjunction or opposition, find by inspection two simultaneous longitudes, 
one for each body, that differ by 0° or 180°. The corresponding time 
of the first will be that of conjunction, and of the second of opposition. 

§ 650. But if these longitudes are not to be found in the tables, take 
therefrom two consecutive longitudes for each body, such, that those of 
the first shall differ from those of the second, in order, the least possible. 
Then, denoting the lesser and greater longitudes of the body having the 
greater velocity by V and l'\ those of the other by l 4 and l u respectively, 
and the corresponding times by t' and t", we have, because the longitudes 
of each are given for the same epochs, 

(I" ~ V) - (L - I) : (*" - t') : 



I - V : 



whence 



(i, - n a" - n 



m which x denotes the interval of time from t' to conjunction. And de- 
noting the ephemeris time of conjunction by T e , we have 

{I, - V) (t" - f) 



T e = t' + x = t> + 



('"-0 -('„-',) ' ' 

§ 651. Increasing the- longitudes of one of the bodies by 180°, and 



(177) 



PROJECTION OF A SOLAR ECLIPSE. 



175 



lecting those of the other to differ the least possible from these increased 
longitudes, then will T c become the time of opposition. 

§ 652. T c is the local time on the meridian for which the ephemeris is 
computed. Denoting the longitude, in time, of any other meridian west 
of this one by L, and the local time of conjunction or opposition by T, 
then will 

T=T C -L (178) 



ANGLE OF POSITION. 

§ 653. The angle made by a circle of latitude 
with a circle of declination through the centre of 
a body, is called the angle of the body 's position. 

To find this angle, let P be the pole of the 
ecliptic, P' that of the equinoctial, and S the cen- 
tre of the body, and make 

X = 90°- P S = latitude of the body ; 
8 = 90°— P' S = declination of the body ; 
to" = P P / = obliquity of the ecliptic ; 

S = P S P' = angle of position : 

then will 

cos tf = sin X . sin 8 + cos X . cos 8 . cos S. 
and 

cos vt — sin X . sin 8 



Fig. 101. 




cos S = 



cos X . cos 8 
§ 654. If the body be the sun, then will X = 0, and 

COS'S? 



S = 



cos 8 



(179) 



(180) 



PROJECTION OF A SOLAR ECLIPSE. 

§ 655. A solar eclipse can take place only at new moon. Find the 
ephemeris time of the moon's conjunction with the sun. Then, by the 
method of interpolation, determine the sun's true longitude and hourly 
motion in longitude ; the moon's true longitude and latitude, and hourly 
motion in longitude and latitude ; the sun's and moon's horizontal paral- 
laxes, and apparent semi-diameters, and the sun's angle of position. 

§ 656. Conceive a cone tangent to the earth, and of which the vertex 
is at the sun. A section of this cone, by a plane between the earth and 
sun, will give an area upon which the sun's centre will appear tc be pro- 



176 



SPHERICAL ASTRONOMY. 



jected when viewed from different parts of the earth. A section at the 
distance of the moon from the earth, and perpendicular to the axis, is 
called the circle of projection. 

§ 657. The diurnal rotation of the earth carries an observer once around 
his parallel of latitude in 24 hours ; a line connecting him with the centre 
of the sun, describes an entire conical surface in the same time, and a sec- 
tion of this cone by the circle of projection will be the parallactic path of 
the sun as determined by the axial motion of the earth. This ellipse and 
the relative orbit of the moon, with a scale of time on each, indicating the 
simultaneous positions of the sun and moon, being constructed upon the 
plane of projection, all the circumstances of a local solar eclipse may easily 
be predicted. 

Fig. 102. 




§ 658. Sun's Parallactic Path. — Let P G P'Hbe a meridian section 
of the earth by a declination circle through the sun's centre at S ; E the 
earth's centre ; P the elevated pole ; G H the projection of the equator ; 
B A that of the observer's parallel, and NN' that of the circle of projec- 
tion on the plane of the section. The projection A'B', of A B on the 
circle of projection by the lines A S and B S, will be the conjugate, and 
that of the diameter of the parallel, which is perpendicular to A B, the 
transverse axis of the ellipse ; the first beiug in and the second perpendic- 
ular to the declination circle through the sun. 

§ 659. Make 

P = EN'U'= moon's horizontal parallax ; 
« = ES U'=z sun's " " 

I = AE H ' = reduced latitude of place ; 

d = HE S = sun's declination ; 

p = E A = earth's radius ; 

w = number of seconds in radius. 



PROJECTION" OF A SOLAR ECLIPSE. 177 

Draw AC, B D, and F K perpendicular to E S, and we have 

A C = p . sin (I — d) ; B D = p . sin (I + c?) ; 
E C= p.cos(l-d); ED = p . cos (Z + tf). 



Also, Eq. (28), 

/? .<? = n . 

tf ' r P 



#S=p.^; J0Jf=p "; 



whence 

From the figure, 

£ (7 = ^ 5 - ^ C = p . - - p • cos (I - 4 

if 

Then in the triangles S C A and S MA', 

SC : SM :: .4(7 : ^'Jf, 
and by substitution 

1 . cos (/ — d) 

also, 

SD=ES+ ED = P . - + P . cos (Z + <Q ; 

and in the same way as above, from the triangles S D B and S M B\ 

B<M^ f . '*<' + *>— .z^j 

1 + - . cos (I + d) 
u 

But at can never exceed 9", and w is equal to 2 06 264". 8, so that the 
terms into which if — w enters as a factor may be neglected, and we have 

MA r = p.sm(l-d).—~ . . . . . (181) 

if £' = p . sin (I + d) . ^-t- (182) 

From which we see that the length of the projection of any dimension 
at the earth, and parallel to the circle of projection, is found by multiply- 
ing this dimension by {P — ir) — P. 

§ 660, Denoting the conjugate axis A' B r by 2 b, we have 

2b=zMB f -MA r ; 
12 



X78 SPHERICAL ASTRONOMY, 

and by substitution, 



b = p . cos I . sin d 



P -«r 



(183) 



Also, 

FA — p . cos I ; 

and because that diameter of the parallel of latitude, which is perpendic- 
ular to A B, is parallel to the plane of projection, we have, denoting the 
semi-transverse axis of the ellipse by a, 

a = p.cosl. — - — (184) 

And denoting the distance M F' from the centre of the circle of projec- 
tion to that of the ellipse by Y, we have, taking half sum of equations 
(181) and (182) 



Y = p . sin I . cos d 



(185) 



§ 661. Revolve the parallel of latitude about A B till it coincides with 
the meridian section. When the observer is at A, it is to him apparent 
noon ; when at B, apparent midnight; when at 0, the angle OF A is the 
apparent hour angle of the sun, and therefore local apparent time. 

Fig. 102 bis. 




Draw T perpendicular to A B, and S L through the point T. The 

projection of F L will give the distance of the sun from the transverse, 

and that of T his distance from the conjugate axis of his elliptical path. 

Denote the first by y, the second by x, and the hour angle FA by i. 

Then 

FO = FA = p. cos?; 

T = p . cos I . sin h ; 

F T = p . cos I . cos h \ 



PROJECTION OF A SOLAR ECLIPSE. 



179 



and since F L T is sensibly a right angle, the value of E S U, which is 
much greater than E S T, never exceeding 9" ; and because F T L = 
U E P = d,we have 

FZ = FT;$md = p. cos I . cos h . sin d ; 

and projecting i^Z and Of on the circl 3 of projection, there will result 

P -<* 



y = p . cos I . sin d . cos & . 
X = p . cos l . sin A . — — 



(186) 
(187) 



But p h- P is the linear subtense of the unit of arc in which P is ex- 
pressed — say one minute. Calling this distance unity, equations (183), 
(184), (185), (186), and (187) may be written 

6 = cos /.sin d (P' — *') (188) 

a = cos I . (P' — <*') (189) 

T = sin I . cos d . (P' - «') (190) 

y = cos /.sin d. cos h.(P'— <ir') . . . (191) 

x = cos I . sin h . (P" — *') (192) 

§ 662. Let C be the centre of the circle of projection, CiVthe trace 
of a circle of declination through the sun's centre on the plane of projec- 
tion. Take the distance Coequal to F, Eq. (190); through F draw 
A A' perpendicular to CJV, and make FA = FA' = a, Eq. (189) ; take 
F B = F B' —b, Eq. (188) ; and making, successively, h equal to 15°, 
30°, 45°, &a, in Eqs. (191) and (192), construct the corresponding hour 




180 



SPHERICAL ASTRONOMY. 



Fig. 103 bis. 




points of the parallactic path of the sun's centre. The geometrical con- 
struction of this path is indicated in the figure. 

§ 663. Moon's geocentric relative orbit. — Substitute in Eq. (180) the 
values qf ts and 8, and make the angle N C L equal to the resulting value 
of S ; the line C L will be the trace of a circle of latitude on the circle 
of projection. Make CD equal to the moon's latitude at conjunction, 
and draw D E perpendicular to C L and equal to the excess of the moon's 
hourly motion in longitude over that of the sun ; draw EH perpendicular 
to ED and make it equal to the moon's hourly motion in latitude ; 
through H and D draw an indefinite straight line ; this line will represent 
the moon's geocentric relative orbit on the plane of the circle of projection. 

§ 664. Scale of time on the Moon's geocentric relative orbit. — -Make 

m = moon's hourly motion in longitude ; 

n— " " " latitude; 

8 = sun's hourly motion in longitude ; 

i = L D H=. inclination of the geocentric relative orbit to circle of 
latitude through the moon at conjunction. 
m f = moon's hourly motion on relative orbit : 



thei 



cot i = 



(193) 



m' =£ (m — *) . cosec i (\?±} 

From the ephemeris time of conjunction take the longitude of the place 
in time, the remainder will be the local time at which the moon's centre 



PROJECTION OF A SOLAR ECLIPSE, 



181 



is at D. Let e denote its excess above the next preceding whole hour, and 
a the distance from D to the moon's centre at that hour ; then will 

a = (m — s) . cosec i . e (195) 

and laying off this distance from D to the west on the relative orbit, we 
have one point, and the entire hour on the scale of time indicating the 
positions of the moon at different times. From this point lay off distances 
to the west and east equal to the value of m' in Eq. (194), and there will 
result a series of points corresponding to the entire hours, as indicated in the 
figure. In the example before us, the local time of conjunction is about 
2 h .33 p. m., the hour poiut 2 falling about 0.33 of m' to the west of 1). 

§ 665. Parallactic relative orbit of the Moon. — The apparent path of 
the moon in reference to the sun's centre, as seen from the earth's surface, 
is the moon's relative parallactic orbit. To construct this path, draw 
through the sun's places on his parallactic path and the moon's places on 
her geocentric relative orbit, at the same hours, lines respectively parallel 
and perpendicular to C N, Fig. 103 ; a series of rectangles will thus be 
formed : the sides of these rectangles which terminate in the sun's places 
will be the co-ordinates of the moon's parallactic relative orbit, in refer- 
ence to the sun's centre, regarded as fixed. 

Fig. 104 
6k 




For example, draw S Y and MX parallel and S X and M Y perpen- 
dicular to C N\ then making S X and S Y, in Fig. 104, respectively 
equal and parallel to S X and S Y in Fig. 103, and drawing X M and 
Y M respectively parallel and perpendicular to 8 iV, their intersection M 
will give a point in the parallactic relative orbit of the moon. This point 
in the example of the figure corresponds to 3 1 '. Other points being con- 
structed in the same way, the orbit sought and its scale of time may be 
completed. 

§ 666. Beginning, Ending, and Greatest Obscuration. — The hour in- 
tervals on the scale of time being suitably subdivided, with S as a centre 



IS2 



SPHERICAL ASTRONOMY. 





Fig. 104 bis. 






6 


iV 






K 


\ 1 ^^\ 


7^ 


s^ 


K' 




"" " 1 m^ 










Kg? 






;2.5j j 



and radius equal to the sum of the apparent semi-diameters of the sun and 
moon, describe an arc cutting the parallactic relative orbit of the moon in 
the points m and m r ; the corresponding numbers on the scale will be the 
times of beginning and ending, the former being at m and the latter at m t . 
From S draw S m 2 perpendicular to the nearest portion of relative orbit, 
the number at m 2 on the scale will give the time of greatest obscuration. 
Also Q Q ' will be the quantity of the eclipse. 

§ 667. The Angle of First Contact. — With S as centre and radius, 
equal to the sun's apparent semi-diameter, describe the circumference 

V W Z ; with m and m x as centres and radii equal to the moon's apparent 
semi-diameter, describe two other circumferences. The tangential points 
B" and E" will be those of first and last contact respectively. The for- 
mer of these it is important to know in advance to guide the observer's 
attention in his efforts to note the actual time at which the solar eclipse 
begins. The angular distance of this point from the highest point or ver 
tex of the sun, measured around the solar disk, is called the angle of first 
tontact. 

§ 668. To find this angle, transfer the point w, which is at 2\5, to the 
corresponding point S' on the parallactic path of the sun, Fig. 103, and 
draw C S\ This line will represent the trace of a vertical circle through 
the sun's centre on the circle of projection, because every such circle must 
pass through the earth's centre, and therefore contain the point C. Ma- 
king the angle K'SV'm Fig. 104, equal to K' C S'm Fig. 103, the point 

V will be the vertex of the solar disk and VSm the angle of first contact. 
§ 669. By this construction, the beginning of a solar eclipse may be 

predicted within one minute of its actual occurrence, and this will in gen- 
eral be sufficient for the practical purpose of indicating the time to look for 
the instant of first contact — one of the most important elements, as we shall 
presently see. in the determination of terrestrial longitude. 



PROJEJTION OF A LUNAR ECLIPSE. 



183 



For a full and complete investigation of the whole subject of eclipses, 
occultations, and transits, see Appendix XL, which consists entirely of the 
admirable paper of Mr. Woolhouse, first published as an appendix to the 
British Nautical Almanac for 1836. 



PROJECTION OF A LUNAR ECLIPSE. 

§ 670. During a lunar eclipse, the earth's shadow rests, as it were, upon 
the actual surface of the moon, and deprives it of a portion of the solar light 
which it would otherwise receive and reflect to a spectator on the earth. 

§ 671. A section of the earth's shadow at the moon will have the same 
parallax as the moon, both being at the same distance from the earth ; and 
regarding * as appertaining to the centre of this section, P — ir will be 
zero; Y, a, b, x, and y will, equations (188) to (192), be zero, and the 
parallactic path of the earth's shadow on the plane of projection will re- 
duce to a point. 

§ 672. Regarding, therefore, S, in Fig. 104, as the centre of a section 
of the earth's shadow at the moon, and V W Z as its circumference, then 
will m m 2 m, represent, not the parallactic, but the geocentric relative orbit 
of the moon. 

§ 673. Hence, to project a lunar eclipse, find from the ephemeris the 
time of opposition or full moon, the corresponding values of the moon's 
latitude, hourly motion in latitude, longitude, hourly motion in longitude, 
horizontal parallax, and apparent semi -diameter ; also the sun's longitude, 
hourly motion in longitude, horizontal parallax, and apparent semi- 
diameter. 

Fig. 105. 
2V 




§ 674. Then, with S as centre and radius equal to that of tho earth's 
shadow at the moon, Eq. (148), describe the circumference VWZ. Draw 



184 



SPHEEICAL ASTKONOMY. 

Fig. 105 bis. 




S N to represent an arc of a circle of latitude ; make S equal to the 
moon's latitude at opposition and construct the moon's relative geocentric 
path and scale of time, as in § 663-4. With S as centre and radius 
equal to that of the earth's shadow, increased by the moon's apparent 
semi-diameter, describe an arc cutting the relative orbit in B and JE, and 
let fall from S the perpendicular S M on the relative orbit. The numbers 
on the scale of time at B, M, and E will give respectively the time of be- 
ginning, middle, and ending of a lunar eclipse. 




'i 



TIME OF DAY. 

675. The imperfection incident to all machinery makes it impossible 
to construct a time-piece to run accurately to mean solar or sidereal time. 
The best efforts result only in approximations; and when these are made 
to the utmost attainable limits, it must remain for astronomical observa- 
tions and computations to do the rest by detecting and applying from time 
to time the amount of error. 

§ 6*76. Time by Meridian Transits. — When the sun's centre is on the 
meridian it is apparent noon ; and twelve hours or twenty-four hours (ac- 
cording as it is civil or astronomical time), corrected for the equation of 
time, gives the mean solar time at the same instant. When a star is on 
the meridian, its right ascension, in time, is the sidereal time at that in- 
stant. And the most simple and accurate method of finding the time, 
and, therefore, the error of a time-piece, is to note the indications of the 
latter when the west and east limbs of the sun or a star cross the wires of 
a transit instrument properly adjusted to the meridian. A mean of these 
indications gives the watch time of the transit of the sun's centre in the 
first 3ase, or that of the star in the second. The difference between the 
first and the mean solar time gives the error on mean solar time ; and 



TIME OF DAY. 



185 




between the second and the star's right ascension, the error on sidereal 
time. Fig. io6. 

It is not, however, always possible to use 
the transit, and recourse must be had to ob- 
servations off the meridian. 

§ 677. Solar Time by a single Altitude. 
— To find the mean solar time and error of 
time-keeper from one observed altitude of 
the sun : Let P be the pole, Z the zenith, 
and S the place of the sun's centre ; and 
make 

a = true altitude of sun's centre = 90° — Z S 
d = true declination of sun . =90° — PS 
I = latitude of the place . . = 90° — P Z 
P= hour angle ZP£; 
s = error of the watch ; 
T w = watch time of observation ; . 

T a = apparent time of observation ; ^ 

T m = mean time of observation ; 
E = equation of time. 

Then, in the triangle P Z S, we have from spherical trigonometry, 
Sin i P = ± \/ 



/cos (S + I) -cos (S + d) 
cos / . cos d 



in which 



whence 



270° - (a + d + I) 



= 2sin- , /±y 



cos (S + 1).qos{S + d) 



cos / . cos d 



= #*; 



and 



T m =T a ±E=i s P + E 

s=T m -T w = T \P + E 



(196) 

(197) 

(198) 
(199) 



The latitude of the place of observation is supposed known. The value 
of s requires, in addition, the values of a, d, E, and T w to be known. 

To find a and T„, measure with a sextant and artificial horizon, or other 
instrument for taking altitudes, the altitude of the upper or lower limb of 
the sun — say the lower — and note the precise indication of the watch at 
the instant. T„ is thus found, ar. i 






186 SPHERICAL ASTRONOMY. 

a =. observed altitude of the lower limb — refraction -f- apparent semi- 
diameter of the sun -f- the sun's parallax in altitude. 

To find d 1 convert the watch time — supposed not greatly in error, other- 
wise the estimated local time of observation — into the corresponding' local 
time on the meridian for which the ephemeris of the sun is computed, say 
that of Greenwich ; take from the ephemeris the sun's declination for the 
next preceding mean noon, and also the hourly change in declination ; 
then 

' d = declination at the preceding noon db its hourly change multiplied 
into the Greenwich time of observation ; 

the upper sign to be used when the declination is increasing, and the lower 
when decreasing. 

To find 3, take from the ephemeris the equation of time for the next 
preceding mean noon, and the hourly change ; then 

E — equation at the preceding mean noon =h its hourly change into the 
t, Greenwich time of observation. 

§ 6*78. It is usual to avoid the correction for semi-diameter by clamp- 
ing the instrument at some assumed altitude, and noting the time, by the 
watch, that the upper and lower lim.b of the sun attain this altitude. 
The mean of these times will be the time when the sun's centre had this 
same altitude, and it will only be necessary to correct the observed altitude 
for refraction and parallax. 

§ 679. It is to be remembered that P has, Eq. (197), the double sign : 
the positive answers to the case in which the hour angle is west, or the 
observation is made in the afternoon ; and the negative to that in which 
the hour angle is east, or the observation is made in the morning. In this 
latter case, j 1 ^ P must be replaced by 12 h — Jj P if civil, or 24 h — J^ P 
if astronomical time be sought. 

This process for finding the error of a time-piece is called the method of 
single altitudes. 

§ 680. Sidereal time by a single Altitude of a Star. — Proceed exactly 
as in § 677-9, using the declination and observed altitude of the star for 
those of the sun, correcting the altitude for refraction only, and find the 
value of y 1 ^ P, to which add the right ascension of the star as found from 
the catalogue ; the sum will be the sidereal time. 

§ 681. The rules for converting solar into sidereal tame, and the- re- 
verse, together with tables for facilitating the same, are given in the solar 
ephemeris. 



TIME OF DAY. 



187 



§ 682. Time of Sunrise and Sunset. — At the instant of apparent sun- 
rise, the sun's centre is in the horizon ; the apparent altitude of its lower 
limb is equal to minus its apparent semi -diameter, and «, in Eq. (197), 
becomes the difference between the horizontal refraction and parallax. 
Making this substitution in Eq. (197) we find P, and this in Eq. (198) 
gives T m . 

§ 683. Time by Equal Altitudes. — V. the sun retained unchanged his 
declination, equal altitudes would correspond to equal hour angles, and the 
half sum of the watch times, augmented by 6 h when the dial-plate is di- 
vided into 12, and 12 h when divided into 24 hours, would give the watch 
time of apparent noon. Twelve or twenty-four hours, depending upon the 
dial-plate, corrected for the equation of lime would give the mean time of 
apparent noon, and the difference between this and the corresponding 
watch time would give the error. 

But the sun is ever changing his declination, and when the effect of the 
change is to lessen his distance from the elevated pole between the obser- 
vations, his hour angle in the morning will be less than in the afternoon at 
equal altitudes ; the watch time of apparent noon, as above found, would 
be too late, and must be corrected by subtracting therefrom half the excess 
of the evening over the morning hour angle. Conversely, when the effect 
is to augment the distance from the elevated pole, the morning hour angle 
will exceed the evening; the watch time of apparent noon, as found by the 
rule, will be too early, and must be augmented by half the excess of the 
morning over the evening hour angle. 

Denoting this correction by t f1 its value in seconds of time will, Appen- 
dix XII., be given by 

t, = — d . tan d . — f- S . tan I . - = ; 

1440 tan 7 \t 1440 sm 7* 2* 



or making 



^4 

1440 sin n\t f 



1440 tan 7 \t 



= B; 



t, ' = + A . S . tan I - B . <5 . tan d . . . . (200) 
In which 

t = the interval of time between the observations in hours ; 

/ = latitude of place ; 

d .= declination of sun at noon of the day , 






188 SPHERICAL ASTRONOMY. 

6 — double daily variation in declination, or change from noon of 
preceding to noon of following day. 

The value of t will be subtractive for apparent noon, and additive for 
apparent midnight, when 6 is positive, and conversely. 

The logarithms of the values of A and B are given in Table IV., for 
every two minutes, from two hours up to twenty-three ; the latitude of the 
place must be known; the declination is found as in § 677, and 6 is ob- 
tained from the ephemeris. 

§ 684. Change of atmospheric temperature and of pressure. — In what 
precedes it is supposed that when the measured altitudes are equal, the 
true altitudes are so likewise ; but this depends upon the state of the air 
remaining the same between the observations. If the barometer and ther- 
mometer vary, the refraction will vary, and the true altitudes will be un- 
equal when the observed are equal. A further correction becomes there- 
fore necessary, and its value is the increment or decrement of the hour 
angle of the sun, which would change his altitude by a quantity equal to 
the difference between the morning aud evening refraction. The amount 
of this correction, denoted by t u and expressed in seconds of time, is, 
Appendix XIII., given by 

_ (/ - r) . cos a 

y '~ T5 'cos/.cosrf.sinP ^ ' 

in which 

r = morning refraction, in seconds of arc ; 
r' = evening refraction, in seconds of arc ; 
P = half the interval between the observations in arc ; 
a = altitude of sun ; 
d = declination ; 
I = latitude of place. 

This process for finding the time of day, or error of a time-piece, is calfea 
tne method of equal altitudes ; and the value of t /t in Eq. (200), is called 
the equation of equal altitudes. 

§ 685. The altitudes should be taken on or near the prime vertical, 
since in that position the altitudes change most rapidly. 

AZIMUTHS. 

§ 686. In surveys and geodetic operations, it is necessary to determine 
the bearings of objects in reference to the meridian of the station from 
which they are seen. These bearings are measured by the angles at the 



AZIMUTHS. 



189 



mstru- 




zenith included between the vertical circles through the objects and the 
meridian. 

§ 687. True Bearing. — To find the true bearing of an object from a 
given station. Take the instrumental bearing of the object and of some 
heavenly body, and add their ditterence to the true azimuth of the body ; 
the sum will be the true azimuth of the object. 

To find the true azimuth of a heavenly body, note the time it 
mental bearing is taken. 

Let Z be the zenith, P the pole, and S 
the heavenly body, say the sun or a star ; 
make 

P = hour angle ZP S; 
I = latitude of place ; 
d = declination of body ; 
T m = mean solar time of observation ; 
E == corresponding equation of time ; 
T a = apparent solar time of observa- 
tion ; 
T^ = sidereal time of observation ; 
^4© = right ascension of mean sun ; 
A % = right ascension of a star. 

Then with the sun and mean solar time, 
we have 

P = 15 . T a = 15 (T m db E) (202) 
With the mean solar time and star, 

P s= 15 (T m + A Q - A % ) (203) 
With the sidereal time and the sun, 

P = 15 {T % - A Q ±iE) . . (204) 

With the sidereal time and star, 

P * 15 {T m - 4*) (205) 

Also make 

<p = ZP = 90°-/; 
* = PS = 90° -rf; 

A = the angle P Z S s 180 s 
£ = " • ZSP. 




azimuth of the body ; 



190 



SPHERICAL ASTRONOMY. 



Then in the triangle Z P S, from Napier's Analogies, 



tan \ (A + I ) ■= cot \ P . 



tan | (A — I) — cot J P . 



cos 


K* 


-<*) 


COs 

sm 


1(^ 
*■■(* 


-<p) 



. . (203) 

. . (207) 

*« 2 (<5 +9) V 

from which A becomes known. The angle f, or that at the body is tech- 
nically called the angle of variation. 

§ 688. The north star, called Polaris, is often advantageously used for 
this purpose, particularly when it has its greatest eastern or western elon- 
gation. At that time the vertical ciicle through the 
star is tangent to its diurnal path, and its diurnal 
motion will be in altitude alone, and not at all in azi- 
muth, thus affording time for taking a series of azimu- 
th al distances. 

To find the time when Polaris or other circumpolar 
star has its g eatest elongation, observe that the angle 
of variation is at that instant 90°, and in the right-an- 
gled triangle P Z S, right-angled at S, we have 

cos P = tan 5 . cot <p . . 
and 

cos -1 [tan 8 . cot p] . . . . (209) 

sin S 




(208) 



Also 



y. =A + A 



sin A = 



sin <p 



(210) 



§ 689. When the bearing of the sun is taken, the line of collimat'on 
must be directed to one or the other extremity of his horizontal diameter ; 
to the bearing of which must be added the horizontal semi-diameter re- 
duced to the horizon, which is equal to the tabular semi-diameter divided 
by the cosine of ihe sun's altitude. 

§ 690. Variation of (he Compass. — From the foregoing it will be ea«y 
to find the variation of the compass, or, as it is frequently called, the dec- 
lination of the magnetic needle. For this pu pose it will be sufficient to 
take the magnetic bearing of some heavenly body and note the time. 
Then from the time and equations (206) and (207), computing the true 
azimuth, and taking the difference between the magnetic and true azi- 
muths, the problem is solved. 

Or, if the true bearing of any terrestrial object be known, we have only 
to subtract it from the magnetic bearing, as determined by the compass, 
to obtain the same result. 



MERIDIAN PASSAGE, 



101 



§ 691. At sea, or' on land where the ob- 
server is surrounded by prairies or extended 
plains, it is usual to take the magnetic 
bearing of the sun's centre, by observing 
alternately the opposite horizontal limbs at 
the time of rising. Then in the triangle 
Z S P, we have 

Z S = 90° + refraction — parallax = £ ; 
and making 

2 + = f + S + <p, 




. i/shi + -sin (X — -5) , . 

cos i J = y -. — - — V- .... (211) 

sin J . sin 9 

It will be sufficient to regard the horizontal refraction and parallax as 
constant, and the former equal to 33' 45" and the latter to 8" ; thus ma- 
king 

£•== 90° 33' 37". 



MERIDIAN PASSAGE. 

§ 692. It is often desirable to know in advance what will be the indi- 
cation of a sidereal or mean solar time-piece at the instant a given body is 
on the meridian. This indication will measure the hour angle of the ver- 
nal equinox, or of the mean sun at the instant, according as the time- 
keeper is running to sidereal or mean solar time. 

§ 693. Time of Meridian Passage. — To find the local mean solar time 
of a given body coming to the meridian, make 

t = the time required ; 
A' s = right ascension of mean sun at this time ; 
A' = right ascension of the body at the same instant ; 
A s = right ascension of mean sun at Greenwich, mean noon next pre- 
vious ; 
A = right ascension of the body at the same instant ; 
s == hourly change of mean sun in right ascension ; 
m = hourly change of body in right ascension ; 
I = longitude of place in time. 
Then the time at Greenwich corresponding to the 'ocal tima t, will be 
t + / ; and 

A' t = A, + a (*.+ *), 
A' = A + m (t + I) ; 



192 SPHERICAL ASTRONOMY. 

and since all the elements are expressed in time, the difference of right 
ascension of the sun and body, when the latter is on the meridian, must 
equal t\ whence 

A! - A' s = A + m (t + I) — A s — s (t + I) = t ; 
or 

A — A s + I (m — s) 

t = — r-^7— H — ( 212 

1 — (m — s) v 

in which A, A s , m, and s must be expressed in the same unit, say hours. 
§ 694. If the body should be a star, then will m = 0, and 

A — A s — I s , M . 

^ = — ■ (213) 

695. If a planet with retrograde motion, m would change its sign, and 

A — A s — I (m + s) 
t= r-?- -j-^^- 1 (214!) 

1 -+ m + s v ' 

§ 696. If the sidereal, time were asked for, then would A, = 0, 

* = 0, and 

A + Im , x 

t = -~- • (215) 

1 — m v 7 

and if the body be a star, then m = 0, and 

t^A. 

REDUCTION TO THE MERIDIAN. 

§ 697. Some of the most important astronomical determinations de- 
pend upon the measured zenith distances or altitudes of a body when on 
the meridian ; but these measurements it is not always convenient nor 
possible to make, and besides it is desirable to multiply measurements 
as much as possible to secure the advantages of a general average in elim- 
inating errors of observations. The purpose of the next proposition is, 
therefore, to pass from a measured zenith distance or altitude- taken when 
the body is off the meridian to what it would have been had the body 
been on that circle. 

The difference between any two zenith distances, applied with the proper 
sign to either, will give the other ; and when one is the meridian zenith 
distance, this difference is called the reduction to the meridian. 

§ 698. Reduction to the Meridian,— To find the reduction to the me 
ridian. 



REDUCTION TO THE MERIDIAN 



193 




Let P be the pole, Z the zenith, S a 
star, S M an arc of the star's diurnal circle 
cutting the meridian in M, S the arc of 
a horizon 1^,1 circle through the star, and 
cutting the meridian in 0. Make 

x=Z S-ZM = Z - ZM=re- 

duction to meridian ; 
I = latitude of place ; 
d = declination of star ; 
P = hour angle Z P S ; 
z = zenith distance Z S. 

Then because 

P Z = 90° - I ; P S = 90° — d ; 

we have in the triangle P Z S 

cos z = sin I . sin d + cos I . cos d . cos P ; 
but 

cosP = 1 — 2.sin 2 £P; 

and substituting this we get 

cos z = sin Z . sin <£ + cos J . cos c? — 2 cos £ . cos d sin 2 J P, 
== cos (Z — d) — - 2 cos I . cos c? . sin 2 J P. 

But Z M= I — d; and s = # -f I — d ; and therefore, 

cos z = cos x . cos (£ — d) — sin # sin {l — c?) ; 
also, 

cos x = 1 — i z 2 + , &c. ; 

and if the observations be made near the meridian, x will be very small, 
and its powers higher than the second may be neglected. Making this 
supposition, writing the arc for its sine, and substituting the value of 
cos x above, we have 

cos z = (1 — \ a 2 ) . cos (I — d) — x . sin (I -- d). 
Equating these values of cos z, there will result 

A x 2 . cos (I — d) -f x sin (I — d) = 2 cos I . cos d . sin 2 J P . . (216) 

In consequence of the small value of x, it will be sufficient for all prac- 
tical purposes to make an approximate solution of this equation ; for this 
purpose write it 

2 cos I . cos d 



sin (/ — - d) 



. sin 2 ± P— cotan {l — d).\x* 



(217 



13 



194 SPHERICAL ASTRONOMY. 



neglecting the term involving the second power of %, 



2 cos I . cos d ;_ . _ 
x = —r— n -.sm 2 i? (218) 

sm (I — d) * v ' 



and this in Eq. (217) gives 
cos I . cos d 



r. • •> 1 71 , / 7 7\ / C0S ^ ' C0S <A 2 • i -, ^ 

2 s.n= 1 P - cot ( I- d) . ( s - nrF ^ 7 ) • 2 sm'J P, 



sin (£ — d) 
and making, in order to find x in seconds of arc, 

2 sin 2 I P 2 sin 4 i P 

■ *„ — ; m = — r— -^ 

sin 1 sin 1 



, .4 3111 Vj -£ 6 SIU n i . 

fc = ■ '„ ; "» = ■ ', .... (219) 



_ cos Z . cos 0? /cos £ . cos dV . . 

x = h ' ' <i T\ -m.cot(l-d .( , ) . (220) 

sm (l — d) ' \sm(l — d)/ v ' 

§ 699. Now P is to be found from the time when the stai is on the 
meridian and that of observation, being equal to the difference of the two 
converted into arc. These times are to be taken from a time-piece, and 
this never runs accurately to sidereal or mean solar time. If the time- 
keeper run too slow, the difference of its indications would be less than 
the corresponding difference of true hour angles— if too fast, the contraiy ; 
and P, in the formula, must be corrected. 

Let the time-piece lose r seconds a day ; then while the true day will be 
equal to 86400*, the clock indication will be 86400 s — r, and any two 
corresponding hour angles, one being the true and the other that indicated 
by the time-keeper, denoted respectively by P and P \ will bear the re- 
lation 

P : P' : : 86400 : 86400 — r ; 
whence 

86400 = 1 . 

'86400 — r 



86400 
making 

r T = — - — = 0.000011. r (221) 

86400 v ' 

developing the fraction, and neglecting the higher powers of r\ 

P = P' (l + r') = P' + P'r', 

and 

sin JP=:sin \P' cos J r'P' -fcosiP' s i n i r 'P' ; 

making cos | r' P' = 1, squaring and rejecting the term containing the 
second power of sin \ r' P', we find 



TERRESTRIAL LATITUDE. 195 

sin 2 ip = sin 2 iP'+ 2 sin \ P' cos \P' . sin \ r' P' ; 
but 

2sin-iP'.cosiP' = sin P', 

and since P' and r' are both small, 

sin P' = 2 sin A P', 
sinJr'P'zzir'siniP'; 

which substituted above give 

sin 3 i P = sin 2 ji" + 2 r' sin 2 J >' = (1 + 2 r') sin 8 J P' ; 

and finally making 

i= 1 +2 r' = 1 + 0.000022 r . . . . (222) 

and substituting in Eq. (220) we have 

. T cos Z. cos o? _ ., 7X / '.os Z . cos dV , nnn * 

x = i.k. -r-~ -j- - i 2 . m . cot (Z -<Z . ( . .. . ) (223) 

sm(Z — d) y \sm (I — d)J v ' 

in which it will be recollected that r, in the value of i, is the rate of the 
time-keeper, minus when the latter gains and plus when it loses on side- 
real time. 

§ 700, The first term in the second member of Eq. (223) will always 
be sufficient when the observations are made within five or ten minutes of 
the meridian. And it is important to remark, in view of the use presently 
to be made of the value of x, that the latter will not be sensibly affected 
by a small error in the value of Z, and that an approximate latitude may 
therefore be substituted therefor. The values of k and m are computed for 
all values of P'from to 35 m , and inserted in Tables V. and VI. 

TERRESTRIAL LATITUDE AND LONGITUDE. 

§ 701. The determinations of terrestrial latitude and longitude by 
means of astronomical observations and ephemerides, are among the 
most important of the objects of practical astronomy. All appreciate 
the value of these determinations in navigation and geography, and we 
now proceed to consider them in the order named. 

Terrestrial Latitude. 

§ 702. The zenith distance of the pole is always the complement of the 
latitude of the place, and when known the latitude is known from the 
relation 

* - 90° - I, 



196 



SPHERICAL ASTRONOMY. 



the 
the 



zenith distance 
latitude of the 




in which X denotes 
of the pole, and I 
place. 

§ 703. The zenith distance of the pole 
forms one side Z P of a spherical triangle, 
of which the two other sides, Z S and 
P S, form, respectively, the zenith and 
polar distances of some heavenly body, 
of which the angle at the pole is the 
hour angle, or distance of the body from 
the meridian. And the determination of latitude consists in the solu- 
tion of this triangle, the data for this purpose being the true zenith 
distance Z S determined from observation, the polar distance P S found 
from the ephemeris, and the hour angle Z P S, which is always equal to 
the sidereal time of observation, diminished by the body's right ascension 
at the same instant. Having, then, found the true zenith distance by cor- 
recting the observed for refraction, parallax, and semi-diameter when ne- 
cessary, and the body's true hour angle and polar distance from the time 
of observation, the ordinaiy formulas for the solution of spherical triangles 
will do the rest. 

§ 704. Latitude by Meridian Zenith Distance of a Body. — But it is 
desirable, in practice, to select those moments for observations which will 
.give most accurate results, and these are when the hour angle is 0° or 
180° ; in other words, when the body is on or near the meridian, for then 
it has the least change in zenith distance for a given interval of time. 

Make 

z = Z S = true zenith distance of body ; 

d = 90° — P S = the body's declination ; 
P = Z P S = hour angle of the body ; 

A = P Z 5 = 180°— the body's azimuthal angle. 

then in the triangle Z P S, 

cos z = sin I . sin d 4- cos I . cos d . cos P . . . (224) 
sin d — sin I . cos z 4- cos I . sin z . cos A . . . (225 ) 

§ 705. Making P = 0°, the body will be on the meridian some- 
where between the poles on the side of the zenith, and A will be 0° 
or 180°. 

In the first case, the body will be between the zenith and elevated roie 
cos A — 1, and Eq. (225) will become 



wnence 
and 



TERRESTRIAL LATITUDE. 

sm d = sin I . cos z + cos I . sin z — sin (I + 0) , 



19' 



1 = d — z 



(226) 



Fig. 113. 





In the second case, the body will be on the opposite side of the zenith 
from the elevated pole, cos A z= — 1; and if the latitude and declination 
be of the same name, sin d and sin I will have the same sign, and Eq. (225) 
gives 

sin d = sin / . cos z — cos / . sin z = sin (/ — z) ; 

whence 

d = I — z, 
and 

l=d+z 



(227) 



Fig. 115. 



If, in the second case, the declination 
and latitude be not of same name, the 
body will be below the equinoctial ; sin d 
and sin / will have contrary signs, and 
Eq. (225) gives 




whence 
and 



sin (— d) = sin I . cos z — cos I . sin z — sin (I — z) ; 
— d = I — z, 

l^z-d (228) 



198 



SPHERICAL ASTRONOMY. 



If F = 180°, the body will be on the 
meridian below the elevated pole, and 
A = 0° ; cos F.= — 1, and, Eq. (224), 

cos z = sin I . sin d — cos / . cos c?= — cos (l-{-d); 

whence 



Fig. 116. 



and 



z= 180°- (l + d), 




1= 180° -z + d . . (229) 

§ 706. Latitude by Circum-meridian 
Altitudes. — Thus it is easy to find the 

latitude when the meridian zenith distance and declination of a heav 
enly body are known. The declination is found from the ephemeris, 
if the body belong to the solar system, or from the catalogue, if it be a 
star. The meridian zenith distance is best determined by the method of 
circum-meridian altitudes, which consists in measuring with an instrument 
a number of altitudes of the body just before and after its meridian pas- 
sage, noting the corresponding times ; reducing to the meridian, taking an 
average value of the results, and subtracting this from 90°. 

§ 707. Denote by h u h 2 , h 3 , <fcc, the measured altitudes ; r„ r 2 , r 3 , &c, 
the corresponding refractions; p u p 2 , p z , &c, the parallaxes; A the ap- 
parent semi-diameter ; x u x 2 , x 3 , &c, the reductions to the meridian ; n the 
number of observations ; and H the average meridian altitude ; then will 



jj- _ h — r x + p x + x x + h 2 — r 2 + Pi + x 2 + &c . ± ^ 



(230) 



the upper sign corresponding to the lower limb, and vice versa. Denote 
by P„ P 2 , Pz> & c -> the watch hour angler of the body ; that is, the differ- 
ence between the watch time of meridian passage and those of observa- 
tions. These, with tables, give k h k 2 , k 3 , &c, m,, m 2 , m 3 , &c., Eq. (223); 
and making 

2^ = ^,+^ + ^+, <fec. ; 
2 m — m x + m 2 + m 3 +, &c. ; 

2 h = h x -j- h 2 + h 3 -f , &c. ; 

2x = x x -f- x 2 + x 3 + , &c. ; 



Y.h zr Vp 



H= +-^ + t-. 



2 h cos I . cos d 

n ' sin (I— d) 



E»l , „ ,,/COS J. C09C?\2 

'. .cot (*-<*)( ) ±A 

"i \ sm (I — d) / 

(231) 

But this supposes I to be known. An approximate value will, § 700, 

be sufficient ; and to obtain it, correct the altitude nearest the meridian for 



TERRESTRIAL LATITUDE. 199 

refraction, parallax, and semi-diameter ; subtract the result from 90°, and 
substitute the remainder for z in one of the equations (226) to (229) in- 
clusive, according to the case. 

§ TO 8. Latitude by opposite and nearly equal Meridian Zenith Dis- 
tances. — With an approximate latitude, select one or more pairs of stars, 
of which the individuals of each pair shall pass the meridian on opposite 
sides of the zenith, and at nearly equal distances. Then, preserving the 
notation of § 704, writing the subscripts 1 and 2 to distinguish the stars, 
and supposing the declinations to be of the same name as the latitude, we 
have, equations (226) and (227), 

t = d x -f z ly 

I = d 2 — z 2 ; 
and, by addition, 



Denoting by £, and £ 2 the observed zenith distances, and by r, and r, the 
corresponding refractions, we have 

Zi = li + r u z 2 = £ 2 + r 2 ; 

which, substituted above, give 

d, + d 2 Y\ — Z* r, — r 2 

| =-Hr J +7 L jT-+'- L T J • • • • ( 232 ) 

If ^1 = £ 2 > then will r, — r 2 = 0, and we have 

l = d L ±d 1 (233) 



and thus the determination of latitude will be made independent of refrac- 
tion, which is one of the greatest sources of difficulty in practical astronomy. 

If Z, x be not equal to £ 2 , but nearly so, the result may be regarded as 
equally accurate, since the difference of refraction will then be employed, 
which, being very small, will be sensibly free from error. 

§ 709. This simple and elegant method, which is one of the most accu- 
rate, and now very generally used, was first employed by Capt. Andrew 
Talcott, late of the U. S. Engineers. The measurements were made by 
means of a zenith telescope, turning about a vertical axis, and provided 
with a micrometer. The stars were so selected that when one was brought 
within the field of view, and made to thread the micrometer wire as it 
passed the meridian, the other would enter the field on turning the instru- 
ment 180° in azimuth. The second star being- made to thread the wire 




200 SPHERICAL ASTRONOMY. 

by the micrometer motion, the extent of the latter was noted, and gave the 
value of £ f — £ 2 . The value of r x — r 2 was found, of course, from the re- 
fraction tables. 

§ 710. Latitude by Polaris off the Meridian. — The last method we 
shall give is that by Professor Littrow. It consists in observing the alti- 
tude of Polaris out of the meridian, and therefore at any convenient time, 
and reducing, not to the meridian only, but to the pole also ; the data for 
this purpose being the star's polar distance, its true altitude, and corres- 
ponding hour angle. 

Let Z be the zenith, P the pole, S 
the place of the star in its diurnal path 
S S' m, Z S the arc of a vertical circle. 
Make 

I = latitude = altitude of pole = 

90° - ZP\ 
h = true altitude of star = 90° — Z S\ 
P= Z P S = hour angle of star ; 
4* = reduction to the pole = Z P — Z S; 
A = P S = star's polar distance. 

Then 

± = h-l; 
and 

I s= h — 4' ; 

so that the latitude is known when 4> is known. 
In the triangle Z P S, we have 

cos Z S = cos P S . cos Z P + sin P S . sin Z P . cos P; 
and replacing the sides by their values in terms of A, h, and I, or k — 4> f 

sin h = cos A . sin (A — 4<) -f- sin A . cos (h — 40 . cos P , 

dividing by sin h and factoring, 

1 = cos ^ . (cos A + fi i n A eot h . eos P) — sin ip . (cos A . cot h — sin A . cos P). 

Make 

a = cos A + sin A . cot k . cos P, V (q*A\ 

b = eos A . cot h — sin A . cos P ; ) 

and tht above may be written, 

1 = a cos >L — b . sin ^ (235) 

Now, A is a small angle, not more than 1° 40' ; and replacing cos A and 



TERRESTRIAL LATITUDE. 201 

wii A by their values in terms of A, equations (234) become, omitting the 
powers of A above the third, 

a = 1 — \ A 2 + (A — \ A 3 ) . cot h . cos P, 
b = (1 - J A 2 ) . cot h - (A -{A 3 ) cos P. 
Let 

^ = AA + Ba* + (7a 3 +, &g (236) 

be the development of 4* according to the ascending powers of A, in which 

there can be no independent term ; since, when A = 0, then will 4 = 0. 

Whence 

cos 4 = 1 — \ A 2 A 2 — A B A 3 , 

sin 4 = A A + B A 2 + (C- jrA') A 3 . 

Substituting the values of a, b, cos -4, and sin 4? m Eq. (235), we have the 

identical equations, 

cot h . cos P — A . cot h = 0, 

- J (1 + A 2 ) + A cos P — B cot h = 0, 

± A - i v l-f 3^ 2 )cos P - (C - \ A z ) = 0. 

Whence 

^4 = cos P ; 

^ = — A sin 2 P . tan A ; 

(7 = J cos P . sin 2 P ; 

which in Eq. (236) give 

4 = A . cos P - \ sin 2 P . tan U . A 2 - + ± cos P . sin 2 P . A 3 . 

To express 4 aQ d A in seconds, write 4 sm 1" f° r 4 an( ^ ^ sm 1" f° r 
A, and make 

m = -1 sin 1", ?i = J sin 2 1", 
then will 

4- = A cos P — m (A . sin P) 2 . tan h + n . (A . cos P) . (A . sin Pf (237) 

This value applied with its proper sign to the observed altitude, cor- 
rected for refraction, will give the latitude. It is best to take some half 
dozen altitudes, and to note the corresponding times in pretty rapid suc- 
cession ; a mean of the altitudes corrected for refraction will give h, and a 
mean of the sidereal times diminished by the right ascension of the star, 
and the remainder multiplied by 15, will give P. 

§ 711. This method is of such practical utility as to have caused the 
insertion into *he English x\stronomical Ephemeris and Nautical Almanac 
of three tables, of which the first contains the value of A cos P for every 
10 minutes, sidereal time, for a mean and constant value of A; the second 
contains the values of — m . (A . sin P) 2 . tan h ; and the third contains 



202 SPHERICAL ASTRONOMY. 

corrections to be applied to the values in the second tible. The secona 
and third tables are arranged in the form of double entry, the arguments 
for the former being the sidereal time and altitude, and in the latter side- 
real time and date. 

The third term of Eq. (237) is neglected as being insignificant. 

Longitude. 

§ 712. The longitude of a place is the angle made by its meridian with 
some assumed meridian taken as an origin of reference. The problem ot 
longitude is much more complex than that of latitude, and its solution 
consists, as we have seen, § 94, in finding the difference of local times 
that exist simultaneously on the required and first meridian. 

§ 713. Longitude by Chronometers. — Could the motion of a time-piece 
be made perfectly uniform, and the angular velocity of its hour-hand equal 
to that of the earth's axial rotation, without the risk of variation, the de- 
termination of longitude would be a simple matter. It would then only 
be necessary to put the time-keeper in motion ; on a given meridian ascer- 
tain, by the methods explained, its error on the local time of this meridian ; 
transport it to the unknown meridian, determine its error on local time 
there, and take the difference of these errors ; this difference would be the 
difference of longitude of the meridians in time. 

But such time-pieces cannot be made. The results to which they would 
lead, may, however, be approached within limits all-sufficient for practical 
purposes. It is only necessary that the time-keeper shall run uniformly, 
a condition which chronometers have been made so nearly to attain as to 
vary their rate but half a second in 31536000 seconds. 

§ 714. By daily observations find the error of a chronometer ; from the 
variation of the error during the intervals between the observations, find 
that for 24 chronometer hours. This will be the rate. Make 

e = error on local time on given meridian, at some given epoch ; 

plus when too slow, minus when too fast; 
e = error on local time on required meridian, atsome subsequent epoch ; 
e n = error on local time on given meridian, at this last epoch ; 
r = rate; minus when gaining, plus when losing; 
i = interval of chronometer time between the epochs at which e and 

e t are found — always plus ; 
I — difference of longitude. 
Then I — e l — e u ; 

e u = e + ir ; 



TERRESTRIAL LONGITUDE. 



203 



whence I = e t — e -\- i .r (238) 

§ 1\o. Longitude by Lunar Distances. — The moon has a rapid motion 
in longitude. Her geocentric angular distances from the sun, planets, and 
fixed stars that lie in and about her path through the heavens, are com- 
puted in advance and inserted into the Nautical Almanac. From these 
hours and distances is readily found, by interpolation, the Greenwich time 
corresponding to any given distance not in the Almanac, and the difference 
between this interpolated time and the local time on any other meridian 
at which the moon is found from observation to have this given distance, 
is the longitude of the meridian on which the observation is made. 

§ 716. Measure the altitude of the star, and that of the upper or lower 
bright limb of the moon ; also measure the angular distance from the star 
to the blight li.nb of the moon, and note the local time of this measure- 
ment; correct the altitude of the limb and measured distance for semi- 
diameter ; then correct the altitude of the star for refraction, and that of 
the moon for refraction and parallax. 

Let Z be the zenith, Z S and Z M the arcs 
of vertical circles, the fi.st passing through the 
star S and the second though the moon's cen- 
tre M. The effect of refraction being to ele- 
vate and that of parallax to depress, and the 
patallax of the moon being always greater than 
her refraction, the star will appear at S' above 
its true place, and the moon at M' below her 
true place. 

Make 

h = 90° — Z M' = observed altitude of moon's limb corrected for 

semi-diameter ; 
h' = 90° — Z S' = observed altitude of star ; 
A' = MS' = observed distance corrected for semi-diameter of 

the moon ; 
H = 90° — Z M = true altitude of moon's centre ; 
H' = 90° — Z S = true altitude of star ; 
4 = M S = true or geocentric distance between the moon^a 

centre and the star ; 
z = MZS = angle at Z. 

Then in the triangle M'Z S', 

cos A' — sin h . sin h! 




cos z 



cos h . 20s h' 



204 SPHERICAL ASTRONOMY. 

and in triangle M Z S, 

cos A — sin H . sin H' 
cos z = Yf Wl ; 

cos H . cos H 

equating these values of cos 2, 

cos A' — sin A . sin A' cos A — sin H . sin H 



cos A . cos A' cos H . cos H 

adding unity to both members and reducing, 

cos A' + cos (A + A') _ cos A + cos (H + H) 

cos A . cos A' cos H . cos .£T 

Make 

A + A' + A' = 2 m . , (239) 

whence 

cos (A + A') = cos (2 m — A') ; 

substituting this above and reducing, we find 

2 H+H f . 2 A 

. ., cos - * . sur — 

cos m . cos (wi — A ) 2 2 

cos A . cos A' cos H . cos -£P ' 

whence 

• i 1 / ,, /rr t-t-'n cos iZ~. cos J5T 

sin -i A = y cos^ ±{H-\-H) — . cos m . cos (m — A'), 

J J v y cos A . cos A' v 7 ' 

and making, to adapt the foregoing to logarithmic computation, 



A /cos H . cos JJ 

y ; 77- . cos m . cos M»-A 

cos A . cos A v 7 . 

smp = ccj^+g) • • ( 24 °) 

then will result 

sin i A = cos i (jET + J5T') . cos 9 . . . . (241) 

§ VI 7. The quantities A, A', -£T, H, and A', are obtained from observa- 
tions, and the corrections for semi-diameter, refraction, and parallax applied 
thereto; the value of m is given by Eq. (239) ; the auxiliary arc <p by 
Eq. (240), and, finally, the true distance A by Eq. (241.) This operation 
is technically called clearing the distance. 

§ VI 8. With this distance enter the Nautical Almanac and see if it is 
found therein ; if it is, take the corresponding time from the head of the 
column, and subtract therefrom the local time of observation ; the remain- 
der will be the longitude — west if this remainder be plus, east if it be 
negative. 

§ VI 9. If the precise distance be not found in the Almanac, as it sel 



D = B' + - h . A, + -^TTT^ + &c; 



TERRESTRIAL LONGITUDE. 205 

doin will, find two consecutive distances, one of which is greater and the 
other less. Take these and the next two or more precedirg and following 
distances, and form their first, second, third, &c, differences, denoted re- 
spectively by A h A 2 , A 3 , &c, in which A, is the difference between the 
consecutive distances of which one is less and the other greater than the 
given distance. Make 

D = given distance ; 
D' = nearest distance in ephemeris ; 
T" = ephemeris time corresponding to D\ 
T = Greenwich time corresponding to D : 
t = T —f'. 

Then because the ephemeris intervals are 3 h , will, by the ordinary for- 
mula for interpolation, 

t j_ t (t — 3*v) 

(3 h )\ 2 

supposing the second differences* constant, which we may do without sen- 
sible error, and solving with respect to first power of t, 

t = ^Z-E. 3" (242) 

Neglecting the second difference, we have 

t = ^—^& (243) 

A t 

which in the denominator of the preceding equation gives 

tss: . _j^jr _ 3K ^ ^ ^ ^ 244 j 

and replacing t by its value T — T\ we have finally 

D — D' 

T - p + _3 h . . . (245) 

A. 






§ 720. A single observer begins by taking with his sextant an altitude 
of the star, then an altitude of the moon's bright limb, then the distance be- 
tween the star and moon's limb, then the altitude of the moon's bright limb f 
then the altitude of star, carefully noting the time of taking the distance. 
A mean of the altitudes of the moon and star will give the approximate 
altitudes which the moon and star had when the distance was measured. 



206 SPHERICAL ASTRONOMY. 

§ 721. It is scarcely necessary to add, that if the sun or a planet be 
taken instead of a star, corrections for semi-d.ameter and parallax must 
be added to that of refraction. 

§ 722. Longitude by Lunar Culminations. — If the change in right 
ascension of a point of the moon, in its pas-age from one meridian to 
another, be known, the distance between the meridians becomes known 
from the point's rate of motion in right ascension. Make 

c l — the point's right ascension when on any upper first meridian ; 

c 3 — its right ascension when on an upper known meridian to the west. 

H=z longitude of this known meridian, west. 
I = approximate longitude of any unknown meridian between these. 

L = true longitude of the same. 

€ = L-l 

ct = right ascension of the point when on the upper meridian, of 
which the longitude is L. 

§ 723. — 1st Approximation. Then, were point's motion in right 
ascension uniform, 

c 2 — <?! : II : : a — c^ : I 

or H 

I = — - (a - c x ) 

§ 724. — 2c? Approximation. But the moon's motion in right ascen- 
sion is not uniform, and tbe above will in general be erroneous, and by 
the quantity e, which is a small arc of longitude ; and we have 

L = I + e 

The arc e being small, the moon's motion in right ascension will be 

sensibly uniform while between the meridians, through its extremities. 

Make 
a x =. the lunar point's right ascension when on the meridian, of which 

the longitude is I, 
v = the point's rate of motion in right ascension ; and let this be 
measured by the distance in right ascension over which the 
point would move, with this rate constant, while between 
the meridians of which the distance apart is //. 

Then by the principle above, writing e for I, v for c 2 •- c xi and a x for c u 

we have jj 

e = — • (a — a x ); 
v 

and this in the abov i gives 

L = I -| • ( a— a,) 

v 



TERRESTRIAL LONGITUDE. 



207 



§ 725. Now, these equations will be equally true from whatever 
point of the equinoctial, taken as an origin, the right ascension be 
estimated. For convenience, take the origin at the declination circle 
through the lunar point at its last passage over the first, or meridian of 
the Ephemeris. Then will 



c 2 = change of right ascension between the known 

meridians, 
a = increase of right ascension from the first to 

the intermediate or required meridian, 
oij = increase of right ascension from the first to 

the approximate meridian I. 

With this new notation the above equations become 



1 B 

I = — a 

L = I H (a 



«i) 



(246) 
(247) 



§ 726. In the Nautical Almanac and Astronomical Ephemeris are 
given the right ascension of the point of the bright limb at which a 
declination circle is tangent to the lunar disc, and also the right ascen- 
sions of one or more stars, at the instant of passing the upper and lower 
meridian of Greenwich for every day in the year. The stars are so 
situated as to lie about the moon's parallel of declination, and not far 
from her in right ascension. 

§ 727. — 1. Interpolation. Take the following scheme : 



I 


F 


*i 


*2 


*3 


\ 


*5 


t'" 


a'" 


W 










t" 


a" 


V 


c" 


d' 






t' 


a' 


b 


c' 


d 


e' 


f 


t, 


a, 


h 


C , 


d, 


e , 




t„ 


«// 


K 


C u 








hu 


«/// 













in which the column I contains the independent variable, or argument, 
as time, terrestrial longitude, degrees, and the like ; F the value of a 



208 



SPHERICAL ASTRONOMY. 



function of this variable, as found in any set of tables ; A w A 2 , A 3 , etc., 
the first, second, third, etc., orders of differences of these functions. 



the interpolated value of the function correspond- 
ing to any given value t t of the argument be- 
tween t' and t t ; 



A 3 = d, 

A -±±*1 



Then, limiting the operation to the fourth order of differences, will 

s = a' + A t + Bt 2 + Ct 3 + D t + 
a x = s - a' = At + Bt 2 + Ct \-JDt* 
in which 



(248) 



(249) 



(250) 



(251) 



A = A, - 1 A 2 + ft A 3 + ftA v 1 

c = 4 a 3-tV a « 

^ = A^4. J 

Also taking first differential coefficient of the function (249) 

v =z A + 2£t+ 3Ct 2 + ±I)t 3 . . . 

which would be the increment of the function for an increment ot-t 
equal to unity, were the function to increase uniformly and at the rate 
it had for any arbitrary value for t s . 

§ 728. — 2, Observations. Make 

.<?£ = sidereal time of moon's bright limb passing meridian. 
A B — clock time of same passing line of colimation. 
e$ = clock error at same instant. 

i§ =z error of transit for altitude of moon, in time seconds. 
Then, §§211 and 729, 



TERRESTKIAL LONGITUDE. 209 

s B = h& + e& dc (1 =p cos £.sec D . p .sin P): 

» s ^ & 2 _ o,04166. m v ^ Y » 

the upper sign before meridian passage, and in which I is the latitude 
of the observer, p the radius of the earth at his place, D the moon's 
declination, P her equatorial horizontal parallax, and m her daily mo- 
tion in right ascension, in hours. 
Making similar notation for a star, 

s^ = A^ + 6* + i$\ 

subtracting this from the preceding, and writing <p for e a — e%, the 
clock acceleration in the interval, in time, between the moon and star, 

ss>— s*=h$— h* + q>±- — -— ^r~— (0,04166m=pcos?.secJ[>.p.sinP) 

On a second meridian, to the west, a similar equation is found, with the 
variables accented. Taking the difference, and making 

«,= ±| - — g — — £— ,004 16,6. m qr— ^- — — ~ -cos J.p.sin P) 

\1 — 0,04166 ' ^ 1 — 0,04166. m Y J 

there will result 

A = «' B -« B = (A , & -A , s|l + <p ; )-(A 9 -A l|c + <p)±* 1 . . (252) 

Or, if there be but one observer with Ephemeris, then will i§ = Q, and, 
omitting accents, the value of k 2 becomes, 

*, = ± 1 _ ' \ u6m ■ (0,04166 . m =p see B . cos i . p . sin P) r 

»9, is found by the method of 13, Appendix II., p. 261. 

in which A denotes the same as a, in Eq. (246), when a single observer, 
on. an unknown meridian, employs the ephemeris elements, as given foi 
the next preceding passage over the first meridian, with those of his 
observations, to get the increase of right ascension requisite to find his 
approximate longitude I; and the same as a — a v in Eq. (247), when 
he employs either the interpolated or observed increase of right 
ascension fur the meridian of which the approximate longitude is I, 
to correct its place. In the first case c 2 is the difference of the point's 
right ascension, as given for the next preceding upper and next 
following lower culmination over the first meridian ; in the second case 
v will be given by Eq. (251); and in both, H will be 12 hours of 
longitude. 

14 



210 



SPHERICAL ASTRONOMY. 



Example, 
Obseevations. — West Point, i845, Feb. 18. 
f Geminorum . . 6 h 54 n »4i s , 75 
<J " .. 7 10 38,97 

j> W. Limb 7^38»o6»,76 

f Cancri . . . . 8 o3 06, 11 

3)22 08 26 , 83 7 22 48 , 94 ... & l5» I7», 81 

ki = o ; Clock rate, + 3 s — o , o3 

Nautical Almanac. — Greenwich, same date. 
£ Geminorum . . 6 h 54 m 57 s , 41 
d " . . 7 10 54,36 

J W. Limb 7 h 27m 4^ 66 

£ Cancri . . . . 8 o3 21 ,44 

3)22 09 i3 , 21 7 23 04 , 40 . . . o 04 43 , 26 

a = A = . . . 0* io»34 s , 53 

Then, Eq. (246), 

H — i2koo™oo«, Log . . 4,6354837 

a — 00 10 34 , 53 " . . 2 , 8024520 

Nautical Almanac c a = 00 25 41,18 " a. c. 6,8121918 

I = 4 56 28, " . . 4, 2501275 

Next, interpolate change of right ascension for I; 

4 h56m 2 8», Log . . 4,2501275 

t, - t' 4 h 56» 28s „ _ ,, e M 

t = - = ; 12 00 00, " a. c. 5 , 3645 1 63 

t . . . . " . . -9,6146436 

Nautical Almanac. 

Feb. 17, L. C. 7 h oi m 56 s , 27 

25» 5i s , 39 
" 18, TJ. C. 7 27 47 , 66 ( - io», 21 

A x = 25 41 , 18 A 9 = £2 «{ A, = — o», a* 

" « L. C. 7 53 28 , 84 f - 10 , 46 

25 30,72 
*« 19, U.C. 8 18 59,56 

A = 25«> 41 s , j8 + o5», 17 — o s , 02 = 25 m 46 s , 33 
B-— 05,77+00,06 =— o5 , n 
€ = — 00 , 04 



TERRESTRIAL LONGITUDE. 2M 

Then, Eq. (249), 



A . . 
t . . 


Log 
« 

Log 
u 

Log 
« 


. . 3 , 1893022 
. . 9 , 6i46438 


Nos , . 

Nus . . 
Nos . . 




B . . 


2 , 8039460 

. . , 7084209 
. . 9 , 2292876 


636s, 72 


C . . 


9 » 9 3 77°85 

. . 8 , 6190933 
. . 8 , 8439314 


.- , 87 




7 , 4630247 
lo» 34 s , 53 = a 


— o,oo3 




635,85 
634 , 53 



«—«,= .... — i», 32 
Again, Eq. (251), 

A . . . . Nos . . 25» 46», 33 

B . . Log . . o , 7084209 

t . . " . . 9 , 6i4«438 

a . . u . . o , 3oio3oo 



0,6240947 Nos . . — 4,21 



G . . Log . . 8 , 6190933 
t* . . " . . 9 , 2292876 
3 . . " . . o,477«2i3 



8 , 3255022 Nos . . — o , 02 

= . , . . 25" 4a J , 10 

Then, last term of Eq. (247), 

E . . Log . . 4,6354837 

a — a, . . " - . . o, 1205739 

v . . " a. c. 6,8118875 

= — 36», 97 . . " . . 1 , 5679451 Nos . . 00 36 , 97 

Eq. (247), 

L = 4 h 56™ 28» ■- 36«, 97 = 4 h 55«n 5i*, o3 



212 SPHERICAL ASTRONOMY. 

§ 729. It frequently happens that the moon cannot be observed on the 
middle wire, in which case she is far enough from the meridian to have a 
sensible parallax in right ascension ; and as it may be very desirable not to 
lose the observation, this parallax must be computed and applied tc the 
apparent hour angle from the middle wire, which is supposed to be nearly 
coincident with the meridian. 

Denoting the hour angle by h, the parallax in hour angle by A //., th • 
geocentric latitude by I, the moon's declination by D, and her horizontal 
parallax by P, then, Appendix XL, p. 379, 

A h = p . cos I . sin P . sin h . sec D ; 

and to make this applicable to the case before us, h will denote the equa- 
torial interval, in sidereal time, from the lateral to the central wire. This 
angle being small, its arc, expressed in seconds of time, may be taken foi 
its sine, in which case, Ah will be in time-seconds, and the true distance 
of the moon's limb from the central wire, denoted by k t , will be 

K, == k . (1 — p '■'. cos / . sin P . sec D) ; 

and the reduction to the meridian, denoted by r, in time-seconds, 

h 1 — p . cos I . sin P. sec D 
cosZ> ' T— 0,04166 7m * 

in which in is the moon's daily motion in right ascension in hours. 
The upper sign, when the observation is before the middle wire. The 
quantities p and I are found from tables on pp. 336, 337. 

§ 730. It also often happens that two observers do not, use the same 
number of wires, or if they do, that the same stars are not observed at 
the same number. Such observations are not of equal. weight. To rind 
.the relative value with which such observations should enter into the 
ifinal determination, Professor Gauss has given the following formula, 
deduced from the principle of least squares. 

Let the number of wires on which the moon is observed at one place be 
denoted by n, and at the other by n ; and let the number of wires at 
which the stars are observed at the first place be a, b, c, &c, and at the 
other be a', b\ c', &z. Make 

= X ....... . (253) 



aa 



n -\- n 
bb' 



; = & -^=7, fcc. v .. (254) 



y 4 fe • (2-oo) 



TERRESTRIAL LONGITUDE. 21$ 

Then, if W denote the weight of each day's comparison, will 

. ^V- • • • • • • (^ 

in which z is the same as " — ' in Eq. (247) ; and for the weight of the 
result of all the comparisons, we have 

2W=2 . ^ 8 " . . .... (257) 

(tf + X) z 2 v ' 

in which 2 expresses the sum. 

, Let e denote the probable error of observation, and E the probable error 

of the final result, then will , 

E = 6 -' ....... (258) 

(f x- ■ ■ '■ 



ifi 



(* + *)«■ 



§ 731. Longitude by Telegraph. — One of the simplest and most; 
accurate methods for finding differences of longitude, is to telegraph to 
a western, the instant of a fixed star's culmination at an eastern station, 
and, conversely, to telegraph to the eastern the instant of culmination 
of the same star at the western station. The local times of both events 
being noted, the difference, as recorded at the same station, corrected 
for rate of time-keeper, gives the difference of longitude. 

The instant of culmination of the moon's bright limb being also sig- 
nalized in the same way, the difference of time, as recorded at the same 
station, corrected for rate, as before, gives the difference of longitude 
augmented by the limb's change in right ascension during the interval, 
and the excess of this interval over that for the fixed stars is the 
change itself. Thus the telegraph, where it connects stations remote, 
froir one another, gives the means for finding differences of longitude 
and for correcting the lunar ephemeris, and, therefore, the elements 
employed in the method of lunar culminations, for use at stations 
having no telegraphic connections. 

§ 732. Longitude by Solar Eclipse, or by Occultation. — The follow- 
ing elegant and accurate solution of this most important problem is, 
in substance, due to Mr. Woolhouse ; it first appeared in the NauticaJ 
Almanac for 1837. 



214 



SPHERICAL ASTEONOMY. 




Let M and S, be the moon and sun, in such geocentric Fig. im 

positions as to appear in external tangential contact 
to an observer on the earth's surface ; the local time 
of this observer will be that of beginning or ending of 
the local eclipse. Conceive a fictitious sun, s, at the 
distance of the moon, within and tangent to the visual 
cone that projects the true sun on the celestial sphere 
for this observer. This fictitious sun will be in con- 
tact with the moon ; and any parallactic effect on the 
one, due to a change in the observer's place, will be 
equal to that on the other. Transport the observer 
to the centre of the earth ; the moon and fictitious sun will appear to 
shift their places with respect to the true sun ; but, being in actual, 
will remain in apparent contact. The apparent disk of the fictitious 
sun and of the moon will diminish ; and the size and place of the latter 
will become those of the ephemeris at the instant of observation. The 
change of the fictitious sun's place, in reference to that of the true sun, 
wiil be the effect of relative parallax. Apply this parallax to the 
place of the true sun. and diminish his disk by a quantity equal to the 
diminution of the fictitious sun ; the result will be the place and size 
of the latter body in apparent contact with the moon, to the observer 
at the central station. The ephemeris time of this contact, diminished 
by the local time of observation, will give the longitude of the observer. 

Thus, the determination of terrestrial longitude, by a solar eclipse, is 
reduced to finding the ephemeris time when the true disk of the moon 
comes in contact with a disk of a given size, placed at a given place. 
The principle is the same for an occultation of a star by the moon. 

In the case of a solar eclipse, the apparent time of observation, 
converted into arc, gives the hour angle of the sun's centre at that 
instant ; and, as the declination of the sun is never subject to a very 
rapid daily variation, this element may be taken from the ephemeris, 
with sufficient accuracy, for the approximate local time on the meridian 
for which the ephemeris is constructed, deduced from an estimated 
longitude, or rough longitude, by account. 



Take 



a = right ascension, 

h = hour angle, 

8 = declination, 

cr = apparent semi-diameter, 



1 



of true sun ; 



TERRESTRIAL LONGITUDE. 215 

Take also 

a = right ascension, ~) 

S = declination, \- of fictitious sun, 

<f = apparent semi-diameter, J 
Act, =- Ah = relative parallax of moon in right ascension, 
Ad = " " " declination, 

A<f = diminution of fictitious sun's semi-diameter; 

Then will 

a = a -f- Ah, 
d a = S + AS, 

<f Q = a — Ac. 

Let N, be the north pole; M, the place of the 
moon at the instant of contact; m, her place when in 
conjunction with the fictitious sun, s. 

Make 

( t ) = any convenient ephemeris time, near this conjunction, 
(^4) .= moon's right ascension at (t), 
(D) = " declination at (t), 
A x = " relative motion in right ascension at (t), 
D x = *' " " declination at (t), 

t = time of true conjunction with fictitious sun, s, 
D = declination of the point m at this time. 

Then, employing the parenthesis to indicate the values of the several 
quantities at the time (t), we have 

(«„) = («)+aA, «. = («) + (ao) 7 ( ^ , 

(*.) = (S) +A5, D Q = (D) t fe) ~ {A) n v 

--1 




216 

Now, make 



SPHERICAL ASTRONOMY. 

h = m s = D Q — (<5 ), 

A = Ms, 
90° 4 *l = ifms, 
90° +4, = sJf»i/ 



then will 



tan v\ = — 



A 



A 1 cos (Z>)' 




and, by the triangle m Ms, considered as plane, 



, h • cos y\ 
cos Np = ; 



and, from the spherical triangle N M ' s, 



sin MNs = 



. sin (ri -f 4,) 

— sin a —rjvr 1 ; 

cos {D) 



or as the small arcs are proportional to their sines, 



MNs = — A 



sin (rj -f 4,) 
cos (D) 



And the time required for the moon to change her hour angle by this 
quantity, will be 



MNs_ A sin (v\ 4- jj 

^ x ~ ~ A^ cos (I)) ' 



TERRESTRIAL LONGITUDE. 



217 



which, subtracted from t Qi will give the ephemeris time of observation. 
Denote this time by t, and we have 



, = w + W-rid) + AV-=^a (259) 



The longitude, from the meridian of the ephemeris, is found by the 
difference between this time and that of observation, previously making 
both apparent, or both mean time, by applying the equation of time; 
and it will be west or east, according as the ephemeris time is greater 
or less than that of observation. 

To find Aa, and A8, take Eqs. (2), Appexidix XI., p. 379, and write 
therein Aa for Ah, P for sin P, A8 for AD, 8 for D and D f , unity for 
cos \Ah, and substitute for h its value h' — Ah; we find 



r> COS I . . 

Aa =z p ' P ' - — — = • sm n. 

v cos 8 l (260) 

AS = p • P • (sin / • cos 8 — cos I • sin £ cos (h — -^Aa) ; 



in which I denotes the central latitude ; and, employing the method of 
solution in Appendix XI, page 381, we have 



-rt cos I . _ 

Aa = p • P • - • sin rt, 

cos o 



(h) 



iAa, 



tan & = cos (^,) • cot /, 



tan M = 



sin 



tan s = tan (d + £) • cos J!^ 
A# = p • P • cos M- cos s. 



cos (d + 8) 



tan ^, 



(261) 



To find AC, resume Eq. (27), substituting therein o for s, <f r for «', 
cos (90° — s) for cos Z, unity for cos z ; and we have 



fl" 



* w — p • P • sin e ' 



218 SPHERICAL ASTRONOMY. 

subtracting unity from both members, clearing the fraction, writing 
P — ie for P, and then P' for p (P — tf), we have 



tf • P' • sin s <S P' 100 • sin s 

tf' — ft = AC = ■ 



P' • sin s 10 10 w — P' • sin s 



Taking the average value of P' in the denominator, say 57' 03", 5, 
and p = 1 ; and, expressing ft and P' in minutes, in which case w 
== 3437, 45, we may write 



Atf = To'To* /; 



in which Atf will be expressed in seconds, if 
100 X 60 • sin g 



/ = 



3437', 45 — 57', 06 • sin s 



For an occultation of a star by the moon, the calculation will, in 
some respects, be slightly abridged. The characters A l and D x 
become the absolute motions of the moon in right ascension and 
declination ; the semi-diameter tf, and its diminution AC, will reduce to 
zero; and the angle s, which is only used to get Arf, may be dispensed 
with; in which case it may be better to employ Eqs. (260) than 
Eq. (261) ; or Eqs. (261) may be modified into the following con- 
venient expressions, by eliminating M and s ; viz. : 



Aa = p • P- — • sin h, (k) = h — jAa, 

COS Oq 

/n , j. ^ . 7 cos (6 + 8) 

tan 6 = cos (h) ■ cot /, Ad = p • P- sin / * — — - 1 - 

v ; ' v cos 6 



(262) 



It will be useful here to recapitulate the equations in a form suited 
to the facilitating of arithmetical calculation, and separately to arrange 
them for an eclipse of the sun, and an occultation of a star by the 
moon, to preserve distinctness. 



TERRESTRIAL LONGITUDE. 



219 



I. — Eclipse of the Sun. 

1. With the longitude by account find the corresj onding Greenwich 
time, and thence from the ephemeris take out the sun's right ascension a, 
declination 5, and semi-diameter ft ; the horizontal parallaxes P, it ; also 
take out the moon's declination D roughly to the minute. 

Reduce the latitude by the table on p. 336, and with p from the table 
on p. 337, Ap. XI, find 

P> = ? (P-«)- 
h = apparent time of observation reduced into arc. 



p = P' cos I sin h ; Jh in min. = [7.92082] 7-^; (h) = h — <dh\ 



tan 6 = cos (h) cot / ; 
tan M = - — tan (h) 



cos [d + <J) 



(r = cos (h) cos / ; 
tan s — tan (& + <5) cos J!f ; 



check . 



J8 = B .P' 



JB = cos M cos s ; 

sin d G 



dam time = [8.82391] 



P 



(6 + 6) £ J 



COS £ Q ' 

Jf to be in the same semicircle with A. 



3. With s find the corresponding factor / in 
the annexed table ; then, using P and ft each in 
minutes, 



A,iasoc. = (Q(y./; 



and thence 



f = ft — A ft 
8 = [9.43537] P. 



partial I 

totai or annular 



phase, A = < 





Factor /"for 


t 


diminution of 




©'s semi-diam. 










O-OI . 


10 


o.3i + ' 30 


20 


0.61 " 30 


3o 


0.89 * 28 


4o 


i.i5 - 2b 


5o 


i'3 7 * 22 


60 


i-54 "" 


70 


1.67 ' 13 


80 
90 


i. 7 5, '° 8 
+ • on 



4. Tn the hourly ephemeris of the moon, fix on a convenient time (t) at 
which the moon's right ascension is near to a Q , and *br this time take out 
the right ascension (A) in time, the declination (i '), and their hourly va- 



220 SPHERICAL ASTRONOMY. 

nations ; also the sun's right ascension (a), declination (5), and their hourly 
variations. Then, 

A i = hourly var. (A) — hourly var. (a) in time; 

D { = hourly var. (D) — hourly var. (§) in arc ; 
(a ) = (a) + A a ; 
(§ ) = (d)+A5. 

5. (a n ) — (A) , . r 

m = v o/ v ; ; t Q = (t) + m [3.55630] ; 

/> = (J9)+m.A; k = D -(S ); 

n = [1.17609]^, cos(Z>); 

Di , k cos r\ 

tan t] = ; cos %L = . 

n ' ^ A 

Corresponding- Greenwich mean time = t + [3.55630] - sin (?j q= 4/), 

13 to have a different sign from Z>i : 

upper ) . ( immersion ) . _ . 

> sign when an < . V is observed, 

under ) ■ ( emersion ) 

II. — Occupation of a Sta?' by the Moon. 

6. With the estimated longitude find the corresponding Greenwich time, 
and thence take out the moon's horizontal parallax P, and her declination 
J), roughly to the minute ; also, 

sid. time = apparent time + ©'s right ascension ; or, 
sid. time = mean time + sid. time mean noon, from p. III. of ephemeris ; 
+ accel. on Greenwich mean time ; 
h — sid. time — a, in arc ; 

P' = ?P\ 

a being the star's right ascension. 

V. v 

p= P' coslsmh: Ah in min. = [7.92082] - J -^-\ (h) = h — Ah: 

L J cos D w 

x = P' sin I 00s § ; x' = P' cos I sin 6 cos (A) ; S = 6 + x — - x' ; 

A a in time = [8.82391] ^ ; a = a + A a. 

L J cos o 

8. In the hourly ephemeris of the moon fix on a convenient time (t) at 
which the moon's right ascension is near to a , and for this time take out 



TERRESTRIAL LONGITUDE. 221 

the right ascension (^1), the declination (D), and their hourly variations 
-4„ 2>,. Then, 

m = ^-^-— ; t = (t) + [3.55630] m ; 

2) =_(2>) + m.2>,; £ = i) -5 ; 

w = [l.lT609]^, cos (Z>); 

■A i rA , ni/ln J COS 7J 

tan r) = ; cos -^ = [0.56463] — — . 

P 

Corresponding Greenwich mean time = t + [2.99167] — sin (?) ;p 4,}. 

Practical Rules for Calculating the Longitude from an Observed. 
Occultation. 

With the estimated longitude find the corresponding Greenwich time 
loughly to the minute, and for this time take out from the ephemeris the 
moon's declination roughly to the minute, her horizontal parallax to the 
tenth of a second, and the sun's right ascension in time to the nearest sec- 
ond. To the sun's right ascension add the apparent time of the observa- 
tion, which will give the right ascension of the meridian. The difference 
between this right ascension and that of the star will give the hour angle 
of the star in time, which must be reduced into arc in the usual manner ; 
it will be 

W. 
E. 



}( sreater ) 
when R. A. of meridian is -J * > than R. A. of 



Reduce the latitude of the place by subtracting the correction found in 
the table in Appendix XL, p. 336, for which the nearest correction found 
in the table will be sufficient. 

To the proportional logarithm of the moon's horizontal parallax, add the 
correction answering to the latitude in the following series : 

ooooooooooooooo o 

Lat. . 11 19 24 29 34 38 42 46 50 54 59 64 69 11 90 
Corr. . 1 2 3 4 5 6 1 8 9 10 11 12 13 14 

To the proportional logarithm of the horizontal parallax, so corrected, 
add the log. secant of the reduced latitude and Ihe log. cosecant of the 
hour angle. To the sum (#,) add the log. cosine of the moon's declination 
and the constant log. 0.3010. The result will be the prop. log. of an arc, 
which, subtracted from the hour angle, will give the hour angle corrected. 

To the corrected prop. log. of the horizontal parallax, add the log. secant 



222 SPHERICAL ASTRONOMY. 

of the *'s declination, and the log. cosecant of the reduced latitude. To 
the same log. add the log. cosecant of the *'s declination, the log. secant 
of the reduced latitude, and the log. secant of the hour angle corrected. 
These sums will be the prop. logs, of two arcs. 

The former arc to have the same name as the latitude. 

The latter to have 



( less } 
the dec. when the h. anofle is \ } than 90 c 

( greater J 



a different name from ) .,' , .. , , . (less 

the same name as 



i( west ) 
the * 's R. A., when it is -J [• of the meridian, 

^ east j 



The sum of these two arcs, having regard to their names, will give the 
correction to be applied to the *'s declination to get the declination 
corrected. 

To the sum (5,) add the constant log. 1.1*761, and the log. cosine of 
the *'s declination corrected ; the sum will be the prop. log. of an arc in 
time, to be 

added to 

subtracted from 

to get the *'s right ascension corrected. 

In the hourly ephemeris of the moon, fix on a convenient time at 
which her right ascension is near to that of the star corrected ; and, 
for this time, take out the right ascension, the declination, and their hourly 
variations. 

Subtract the common log. of the difference between the corrected right 
ascension of the star and the right ascension of the moon, from the com- 
mon log. of the hourly motion in right ascension ; to the remainder add 
the constant log. 0.4771 ; to the same remainder add the prop. log. of the 
hourly motion in declination. The former sum will be the prop. log. of a 
time to be 

added to ) , , . . , ~ n . . (greater) - , < ■ 

> the assumed time when % s R. A. is <f V than D's R. A. 

subtracted from \ ( less ) 

to get the time corrected. 

The latter will be the prop. log. of a correction of the D 's declination, 
to be applied with 



the same name as 
a different name from 



>■ hourly var. when Jjc's R. A. is } *? £ than D *s R. A, 

To the common log. of the hourly motion in right ascension, add the 
log. cosine of the D's corrected declination ; and to the sum (S 2 ) add the 
prop. log. of the hourly motion in declination and the constant log. 7.1427. 



TERRESTRIAL LONGITUDE. 223 

The result will be the log. cotangent of the first orbital inclination,* and 
must take 



> hourly motion in dec. when * is < > of D . 



the same name as ) , , x . . , ., . ( north 

a different name from 



To the prop. log. of the difference between the star's declination cor- 
rected and the moon's declination corrected, add the constant log. 9.4354 ? 
and the log. secant of the preceding orbital inclination ; and from the sum 
deduct the prop. log. of the horizontal parallax. The remainder will be 
the log. secant of the second orbital inclination,! which must have the 
name 

S. ) , , . . ( immersion 

__ > when the observation is an < 

Jn. ) ( emersion. 

Add together the two orbital inclinations, having proper regard to their 
names; and to the log. cosecant of this sum add the preceding sum (# 2 ), 
the prop. log. of the horizontal parallax, and the constant log. 8.1844. 
The sum will be the prop. log. of a correction to be applied to the time 
corrected to get the mean time at Greenwich : it must be 



. . } when the sum of the orbital inclinations is 

subtracted 



. (N. 

,s |s. 



By applying the equation of time from p. II. of the ephemeris, there 
will result the Greenwich apparent time, and the difference between it and 
the apparent time of observation will show the longitude of the place from 
Greenwich ; it will be 

' }■ when the Greenwich time is J- f ' ]■ than the observed. 



Examples. 

I. SOLAR ECLIPSE. 

For a solar eclipse, take the example directly calculated in Appendix XL, 
page 41 2 : 

Suppose the beginning of the solar eclipse on May 15, 1836, to be observed to 
take place at i h 36 m 35 s • 6 p. m., apparent time, in latitude 55° 57' 20" N., and 
longitude about 12™ W. 



* With the parallel of declination. f With the moon's limb. 



2M 



SPHERICAL ASTRONOMS 



Here we have 

Observed apparent time 
Longitude .... 


h. 


m. 
36-6 

12. 


= + 


56 m 35 8 .6 


Greenwich apparent time 
Equation of time 

Greenwich mean time . 


i 
i 


48-6 
3- 9 

44-7 


24° 8'- 9 



We hence take from the ephemeris, a =. 3 h 29™ 19 s , 5 = + 18 57' -6, 
= i5' 49" -9, Z> = +i9° 19', P = 54' 24" -4, ff = 8".5, P — - = 54' i5"-<?. 



Latitude + 55° 57' 20" 
Reduction 10 28 



+ 55 46 52 

JT 3.5l267 
9.99902 



P 

P 

P' . 3.5ll69 

cos I 9.75001 



. . . p = 9-99902 

cos (h) + 9 • 96060 . -f" 9 • 96060 
cot I +9-83256 cos I +9-75001 

O ' - 

0+3i 5o-7 tan +9-79316 q. -J-9 . 71061 

<5+i8 5 7 -6 . ■ . — 

______ sm +9.722J1 



o 
AH 



8.9 
6-6 



+ 9-78899 

+9.92162 check +9-92162 
tan (^)+9-64936 

tan M -{-9. 57098 

cos M +9.97180 . +9-97180 
tan (0+(5)+o- 08861 cose +9-81719 

£+48° 58'. 3 tans +o-o6o4i B +9.78899 



sinA+9-6n83 0+<5+5o 48«3 cos +9-80069 B 

p +2.87353 (1) 
cos D 9.97484 

+ 2-89869 
const. 7-92082 

+0-81951 



(A)+a4 2-3 

6+18 57-6 
+ 33-3 A<5 + 33' 18". 4 

<5 +i9 3o-9 cos 9.97430 (2) 



P' 



+3-5ri6 9 
+3-3oo68 



+ 2.89923 (i)-(2) 
const. 8-82391 

. +-i*7-3i4 



A<r 



log. 

A a + O h 0"i 52 s - 86 
a 3 29 19 

a„ 3 3o 12 



i5' 49"-9 

ir -6 P . 3.5i38o 

~~z t: ~ const. 9-43537 

14 49 -6 . 2.94917 



3o 27 .9 



By inspecting the hourly ephemeris of the moon's right ascension on May 15th 
with a = 3 h 3o m 1 2 s . the most eligible time to assume is evidently [t) = 3 h o m o s ; at 
this time we have {A)=3 h 3o m 4- s -84, („ 1 ) = 2 m o s -68, (£>) = + 19 3i' 34"-o, 
(D 1 ) = + 9' 55". 2, (a)z=3 h 29™ 3i s -57, (_,) = + 9^89, (-) = +i8° 58' 21". 4, 
(ii) = + 34 " -8 : with these we proceed as follows : 





m. e. 
. 2 o-68 
9.89 


(A) • 

CO • 


. +9 55-2 
. + 34-8 


A x . 


. 1 5o«79 


I>x . 


. + 9 20-4 



TERRESTRIAL LONGITUDE, 



225 



w 

A a 



h. m. s. 

. 3 29 3i .57 

+ o 52-86 



(i) -f 18 58 21.4 
A<5 + 33 i8-4 



(-0) • 

(A) . 


3 3o 24-43 
3 3o 42-84 


(<y 


+ 


19 3i 39-8 


>o)-M) - 


— 18.41 


A- 
jlog. 


• 




og. ... 

m . 

const, 
j log- - 


— 1 -265o5 
2-o445o 

— 9-22o55 
3-5563o 

— 2.77685 


-f 2- 7 485o (1) 

— 9-22o55 

— 1 -96905 

o° 1' 33"- 1 


t - 

(0 

h + 


Q h 9 m 53s. 2 

3 
2 5o i-8 


W) + 

#o + 
(*o) + 


19 3i 34 -o 
19 3o .9 
19 3i 39 -8 






k 


— 


1 38 .9 






cos (D) 
A, . 

const. . 




9.97428 
2-o445o 
1 • 1 7609 







n . 




3.19487 (2) 


ft . . 


— 19 4r-2 


tan «? . 

cos n . 
k . 

A . 




— 9-55363 (1) 

+ 9. 97 384 

— 1 -99520 

— 1 • 96904 
3-26196 


4> • • 


-f- 92 55-2 


COS <// . 




— 8 • 70708 


I — * • • 


— ii2 36-4 


. sin 

A . 
const. . 




— 9-96528 
3.26196 
3-5563o 




— 6-78354 (3) 


corr. — 


E h 4 m 38 s - 5 


Greenwich mean 
Equation of time. 


— 3.58867 (3) 


t -f corr. -f 


1 45 23 -3 
3 56 -o 


time. 




1 49 19-3 

1 26 35 -6 


Greenwich apparent time. 
Observed ** " 



Longitude 



12 43 « 7 W. of Greenwich. 



15 



226 



SPHERICAL ASTRONOMY. 



H. OCCULTATION OF A STAR. 

Suppose, at Bedford, on January 7, 1836, in latitude 52° 8' 28" K, the immer- 
sion of i Leonis to be observed at io h 39m 22 s - 4 p. m., apparent time, and the 
estimated longitude to be about o h i m W. Required the longitude? 



Apparent time (observation) 
Longitude 

Apparent time (Greenwich 
Equation of time 

Mean time (Greenwich) . 



h. m. 




' » 


IO 39 


Latitude . 


. N. 52 8 28 


O T W. 


Reduc. 


10 57 



10 4o 

7 
10 47 



K 5i 67 3i 
Reduced or geocentric latitude. 



For Jan. 7, at io h 47 ra , we find, from the Ephemeris, ©'s R. A. = I9 h I2 m 4o", 
D's dec. =H". 1 5° 5o', and D 's equ. hor. par. =56' i"'y. 

h. m. s. 

0' 8 R. A 19 12 4o P. L. D's hor. par. . o-5o68 

App. time 10 39 22 corr. for lat. ... 9 

R. A. meridian .... 5 52 2 P. L. corr (l . hor. par . o-5o-jj 

" 5j? 10 23 26 sec. red. lat. . . . o-2io3 

i in time . 4 3i 24 cosec. hour angle . o-o333 

in arc . t>7 5i sum (5]) . . . . o-75i3 

cos. D's dec. . . . 9-9832 

const, log. . . . o-3oio 

corr". . 17 . . P. L. corr". . . . i«o355 
sjc's hour angle E. corr d . . 67 34 



P. L. cori d . hor. par. 


. o-5o77 


• 






• 5077 


sec. s|c's dec. 


. o-oi5o 




cosec. . 


. 


0.5876 


•cosec. red. lat. 


. o-io37 




sec. 




0'2I03 


> " 







sec. corr* 1 . hour 


angle 


0.4184 


N. 42 33- 


P. L. 0-6264 











S. 3 23 -9 






PL... 










sum (Si) . 






corr". . . 1ST. 39 9-1 


o-75i3 


#'sdec. . N. 1 4 58 38-8 






const, log. 


. 


1 -1761 


jjc's dec. corr d . N. i5 37 47-9 






cos. 




9.9836 


corr". . 


O 1 ' 2m 12 s 


• 56 


P. L. corr n . . 


. . 


1 -91 ro 


*'s R. A. . 


IO 23 26 


39 









j|c's R. A. corr d . 10 21 x3 -83 



On referring with the ^c's corrected R. A. to the hourly ephemeris of the moon, 
it will evidently be most convenient to take out the data at n h ; for this time we 
have D 's R. A. = io h 20 m 58 s -47, hourly motion D 's R. A. = 2 m 2 8 '9, D 's dec. = 
W. i5° 47' iJ"«o, hourly motion D's dec. =S. n' 4i"'5. 




TERRESTRIAL^LO^GITUDE 



227 



;jc's com'. R. A. 
D 's R. A. . . 

( diff. . . . 



h. m. s. 
10 21 i 3 -83 

io 20 58'47 

o i5-36 



{ common log . . i ♦ r 864 
com. log. h. m. J) 's R. A. 2-0896 

Remainder . 0.9032 

const, loir o 477i 

h. rn. s. 

corr". . .07 29-9 P. L. 1 • 38o3 

Time assumed 11 o o 











. 


. 


. 


. 





• oo3a 


P. L. 


h. 


m 


.»*s 


dee. 






• 


1 


•i8 7 4 


corr n . 






S. 


, 
1 


27 


•7 


P.L. 


2 


• 0906 


's dec. 






K. 


i5 47 


! 1 


■ o 









Time corr d . 



11 7 29.9 



D \s dec. corr''. N. i5 45 43-3 



com. log. h. m. J) 's R. A, 
cos. J> 's corr 11 . dec. 



2-0896 
9.9834 



sum (S. 2 ) 2-0730 

P. L. h. m. D ? s dec. . . . 1 • 1 874 

const, log 7- J 4*7 

1st Orb. incL K 21° 34' cot. o«4o3i 



2nd Orb. incL S. 6r 
sum . 



S. 39 35 



corr". 
Time corr" 1 . 



h. m. s. 
o 19 44-5 
11 7 29^9 



o ' 

sfc's corr*'. dee. . . K i5 37 47*9 
D's " " . . N. i5 45 43.3 

idiff. (#S. of J>) ~ 7 55-4 

|P,L 1.3563 

const, log. . . . 9.4354 

sec . . . . . o»o3t5 

o-8i32 
P. L. D's hor. par. . o«5o68 

, sec o-3i64 

. cosec 0-1957 

sum (S.) . . . V0730 
P. L. J> s hor. [ ar. . o«5o68 
const, log. . . .8* i844 

P. L 0*9599 



Greenwich moan time 10 47 45 «4 
Equation of time . . 6 3i «o 



r 



Greenwich app. time 
Observed " " 

Longitude 



10 4' i4*4 
10 39 22*4 



5a.o W 



P. S. — The principle of reversing the effect of the relative horizontal parallax 
on the position of the sun, instead of u-^ing the actual effect on the position of the 
moon, may be advantageously employed in the direct calculation of an eclipse for 
a particular place. It will only be necessary to use the parallaxes for the sun 
viewed as an apparent position, and to diminish the semi-diameter by the amount 
derived from the table on page 360. Tim--, it appears, at the beginning of the 
eclipse, for instance, that the contact may be mathematically tested i;i two ways. 
First, we may apply the actual effects of the parallax to the true position of the 
moon, then augment her semi-diamet< r, and thus estab'Uh a contact of the limbs. 
But, if we reverse the operation, and consider the sun to be an apparent body 
under the iufluence of the relative parallax, then clearing it from this supposed 



228 SPHERICAL ASTRONOMY. 

influence by reversing the parallax, and diminishing the semi-diameter, a contact 
will similarly be established with the true limb of the moon ; and this principle, 
in its application to solar eclipses, possesses an advantage similar to that derived 
in the case of an occultation, by considering the star as an apparent place. (See 
Appendix XL, page 399 .)* 

The formulae, Nos. 2, 3, 4, and 5, pp. 406, 407, may, according to this 
method, be supplied by the following : 

2. P' = P (P - ; m = P' cos l\ 

Q x = [9.4180] ; Q 2 = [9.4180] m sin S • 

s =[9.43537] P. 

8 ' * = m 



cosZ>' 

A h in minutes = [7.92082] h sin h ; 
(h) = h — Ah; 
tan 6 = cos (A) cot I ; G = cos (h) cos I ; 

tan M = 7 T~7~x\ tan W i tan s = tan (^ + ^) cos -^"i 

COS 10 —J— O I 



check . 



i? = cos if cos s ; 

sin 6 G 



cos (4 + 5) B ' 
A<5 = i?.P'; 

tf = tf — diminution for s ; 



For|Pf; al , Iphase.A-I*^ 

( total or annular ) ( s ~ <r o . 



k n = r- : A a = k n sin h 



'Q 



COS £ ' 



A a. x = ^j k Q cos A ; A S x = $ 2 sin (A). 

5. # = <5 + A # ; a/ = a — A a ; 

y = (a — A a) cos D ; #, = (a x — A a,) cos D ; 

# = (D + a' corr.) — <5 o ; x x = 2^ — A 5 1# 



* This was inadvertently ascribed to Carlini. Professor Henderson, by whom 
a paper has appeared upon this very point in the Quarterly Journal for 1828, 
page 411, informs me that the method has been long in practice, and that it was 
employed at an early period by Dr. Maskelyne. 



CALENDAR. 229 

§ 734. Longitude by Eclipses of Jupiter's Satellites. — The eclipses of 
Jupiter's satellites are computed in advance, and the times of occurrence 
inserted in the Nautical Almanac, to facilitate the determination of terres- 
trial longitude. After ascertaining, by inspection, about the time an 
eclipse begins and ends, the satellites are watched with a good telescope, 
and the precise local time of entrance into aud departure from the shadow 
noted as nearly as possible. The time given in the Almanac, diminished 
by this observed local time, is the longitude ; west, when the difference is 
positive, east when negative. This method for finding longitude is defec- 
tive, for reasons stated in § 497. 



CALENDAR. 

§ 735. To divide and measure time and to note the occurrence of 
events in a way to give a distinct idea of their order of succession and 
the intervals of time between them, is the purpose of Chronology. 

§ 736. All measurements require standard units. These units are, for 
the most part, purely arbitrary, and are equally convenient in practice. 
But such is not the case in chronology. Time is divided and marked by 
phenomena which are beyond our control, and which indeed regulate our 
wants and occupations. The alternation of day and night forces upon us 
the solar day as a natural unit of time. 

§ 737. To avoid the use of numerous figures in the expression of great 
magnitudes, all measurements must have their scales of large and small 
units, and usually the selection of the larger is as arbitrary as the smaller ; 
but here the phenomena of nature again interpose, and the periodical 
return of the seasons, upon which all the more important arrangements 
and business transactions of life depend, prescribes the tropical year as an- 
other and higher order of unit in chronology. 

§ 738. But the solar day and tropical year are both variable, and are 
therefore wanting in all the essential qualities of standards. Neither are 
they commensurable the one with the other ; they are on this account 
unfit units for the same scale, lu the measurement of space, for instance, 
each unit is constant, and one is an aliquot part of another — a yard is 
equivalent to three feet, a foot to twelve .inches, &c. But a year is no 
exact number of days, nor an integer number and any exact fraction, as a 
third or a fourth, even ; but the surplus is an incommensurable fraction 
which possesses the same kind of inconvenience in the reckoning of time 
that would arise in that of money with gold coins of 101 dimes and odd 
«ents, and a fraction over. For this there would be no remedy but tc 



230 SPHERICAL ASTUONOMY. 

keep an accurate register of the surplus fractions, and when they amount 
to a whole unit, to cast them over to the integer account. To do this in 
the simplest and most convenient manner in the reckoning of time, is the 
object of the calendar. 

§ 739. A calendar is, therefore, a classification of the natural and other 
divisions of time, with such rules for their application' to chronology as 
shall take into account every portion of duration without recording any 
one portion twice. 

These divisions are years, months, weeks, days, and certain periods, to 
be noticed presently, and which are chiefly important in the use made of 
them in fixing upon a common epoch or o;igin of reference. 

§ 740. Julian Calendar. — The years are denominated as years current, 
not as years past, from the midm'ght between the 31st of December and 
1st of January, immediately subsequent to the birth of Christ, according 
to the chronological determination of that event, and this origin is desig- 
nated by the letters A. D. or B.C., accoiding as the year is subsequent or 
previous. Every year whose number is not divisible by four without a re- 
mainder, consists of 365 days, and every year which is so divisible of 366. 
The additional day in every fourth year is called the Intercalary day. 
The years which consist of 36-3 days are called Common years ; those which 
consist of 366 days are called Bissextile, years, and frequently Leap years. 
The mean length of the year by this rule is obviously 365A days, and the 
mode of leckoning t'.me by this unit in the way just described is called the 
Julian Calendar. 

§ 741. The year is divided into 12 months of unequal length. They 
are named, in order of succession, January, February, March, April, May, 
June, July, August, September, Octobe' - , November, and December. 
January, March, May, July, August, October, and December, have each 
31 days ; each of the others except February has 30, and February has in 
a common year 28 and in a bissextile year 29 ; so that the intercalary day 
is added to February. The weeks consist of seven days, named in order, 
Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday. 

§ 742. Gregorian Calendar. — The Julian year consists of 365.25 days; 
the tropical year of 365.24224, making the Julian longer than the tropical 
by 0.00776 of a day, and causing the seasons to begin earlier and earlier 
every year as designated by the Julian dates. In process of time the 
seasons would therefore correspond to opposite dates of the year, and as 
this was likely to interfere with the times of holding certain church fes- 
tivals, Pope Gregory XIII. determined upon a reformation of the Julian 
calendar. 



CALENDAR. 231 

§ 743. In A. D. 325, the seasons, festivals, and Julian dates corres- 
ponded with one another, according to church rule. The reformation was 
effected in 1582. Now (1582 — 325) X O d .00776 = 9 d .6243. Again, 
O ,! .00776 X 400 = 3 d .104. The Pope ordered that the day following, the 
4th of October, 1582, should be called the loth instead of the 5th. This 
brought the date of the sun's entering the vernal equinox to what it was 
in 325, the time of holding the Council of Nice. And to secure this 
coincidence in future, he also ordered that three intercalary days should be 
omitted every four hundred years, the omissions to take place in those 
centennial years which are not divisible by 400 ; so that 1700, 1800, and 
1900, which by the Julian mode of reckoning are bissextile, are made b^ 
the Gregorian common years. There is, therefore, at the present time, 
viz., in the 19th century, a difference of 12 days between the Julian and 
Gregorian dates. The mode of reckoning by the Julian calendar is called 
Old, and that by the Gregorian New Style. New style is followed 
throughout Christendom except in Russia, where the old style is pre- 
served. 

§ 744. Solar Cycle. — This is a period of 28 Julian years, after the lapse 
of which the same days of the week in the Julian system would return to 
the same days of each month throughout the year. For four such years 
consist of 1461 days, which is not a multiple of 7, but 7 times 4 or 28 
years is a multiple of 7. The place in this cycle for any year of A. D. is 
found by adding 9 to the year, dividing by 28, and taking the remainder. 
When there is no remainder, the number sought is 28. 

§ 745. Lunar Cycle. — This is a period of 19 years or 235 lunations which 
differ from 19 Julian years only by about an hour and a half; so that, 
supposing the new moon to happen on the first of January in the first year 
of the lunar cycle, it will happen on that day or within a very short time 
of its beginning or ending again after the lapse of 19 years. The number 
of the year of the lunar cycle is called the golden number, to find which 
add 1 to the number of the year A. D., and take the remainder after 
dividing by 19. If there be no remainder, the golden number will be 19. 
The golden number is used in ecclesiastical dates to determine the civil 
date of Easter. 

§ 746. Cycle of Indiction. — This is a period of 15 years, used in the 
courts of law and in the fiscal organization of the Roman empire, and 
thence^ introduced into legal dates as the golden number into the ecclesias- 
tical. To find the place of any year of A. D. in the cycle of indiction, add 
3, divide by 15, and take the remainder. If there be no remainder, the 
number sought will be 15. 



232 SPHERICAL ASTRONOMY. 

§ 747. Julian Period. — The product of 28, 19, and 15 is 7980. This 
is called the Julian Period ; and it is obvious that after this period, the 
years of the solar, lunar, and indlctiOb ovciea will recur in the same order ; 
that is, each year wil> holci the same place in all the three cycles as the 
corresponding year in the previous period. 

§ 748. As no common factor exists in the numbers 28, 19, and 15, it is 
plain that no two years in the Julian period can agree in its three compo- 
nent cycles, and to specify the number of a year in each of the latter is to 
specify the number of the year in the Julian period, which now embraces 
the entire authentic chronology. The first year of the current Julian period, 
or that of which the number of the three subordinate periods is 1, was the 
year B. C. 4713, and noon of the 1st of January of that year, for the me- 
ridian of Alexandria in Egypt is the chronological epoch to which all his- 
torical eras are most readily referred, by computing the number of integer 
days intervening between it and Alexandria noon of the days which serve 
as the respective epochs of these eras. The meridian of Alexandria ia 
chosen, because it is that to which Ptolemy refers the commencement of the 
era of Nabonassar, the basis of all his calculations. 

§ 749. Given the year of the Julian period, those of the subordinate 
cycles are found as above. Conversely, given the year of the solar, lunar, 
and indiction cycles, to determine the year of the Julian period, proceed as 
follows, viz. : Multiply the number of the year in the solar cycle by 4845, 
in the lunar by 4200, and in the indiction by 6916, and divide the sum of 
the products by 7980, and the remainder will be the year of the Julian 
period sought. 

§ 750. A date, whether of a day or year, always expresses, as before re- 
marked, the day or year current, not elapsed ; and the designation of a year 
by A. D. or B. C. is to be regarded as the name of that year, and not as a 
mere number designating the place of the year in a scale of time. Thus, 
in the date January 5, B. C. 1, January 5th does not mean that 5 days in 
January have elapsed, but that 4 have elapsed, and the 5th is current. 
And B. C. 1, indicates that the first day of the year so named (the first 
current before Christ) preceded the first day of the common era by one 
year. The scale A. D. and B. C. is not continuous ; the year 0, is wanting 
in both parts, so that supposing the common reckoning correct, our Saviour 
was born in the year B. C. 1. 

§ 751. Epact. — The mean age of the moon at the commencement of a 
year is called the epact. It is a name given to the interval of ti'me be- 
tween the first of the year and the next preced ng mean new moon : it is 
expressed in days, hours, minutes, and seconds. Its use is to find the days 



CALENDAR. 233 

of mean new and full moon throughout the year, and thence the dates of 
certain church festivals. 

§ 752. Equinoctial Time. — Astronomical time reckons from noon of 
the current day ; civil, from the preceding midnight. Astronomical and civil 
dates coincide, therefore, only during the first half of the astronomical and 
last half of the civil day. Were this the only cause of discrepancy, it might 
be remedied by shifting the astronomical epoch to coincide with the civil. 
But there is an inconvenience to which both are liable, inherent in the 
nature of the day itself, which is a local phenomenon, and commences at 
different instants of absolute time under different meridians. In conse- 
quence, all astronomical observations require to be given, to render them 
comparable with one another, in addition to their date, the longitude of 
the place of observation from some known meridian. But even this does 
not meet the whole difficulty, for when it is Monday, 1st of January, of any 
year, in one part of the world, it will be Sunday, 31st December, of the 
preceding year, in another part of the world, so long as time is reckoned 
by local hours. 

The equivoque can only be avoided by reckoning time from an epoch 
common to all the earth. Such an epoch is that which marks the passage 
of an imaginary sun having a mean motion equal to that of the true sun, 
through a mean vernal equinox receding uniformly upon the ecliptic with 
a motion equal to the mean motion of the true equinox. Time reckoned 
from this epoch is called equinoctial time. Equinoctial time is therefore 
the mean longitude of the sun converted into time at the rate of 360° to 
the tropical year. 



APPENDIX. 



236 



APPENDIX I. 









X 




< 




. 


B, 


^H 


t— 1 




Q 


M 


Pu 
O 


£ 


h- 1 
Q 


5 


W 




W 


Cu 


Oh 


o 




Ph 




Oh 


<1 




<J 




>3 



*~ o 

o »> 



-. 00 

- co 

o 





S5 


_ 


r^ 


to 


Ci 






o 




-<* 




CNf 


lO 


fa 


hi 




o 


CO 


o 


cj 


o 






o* 


rf 


CO 


c* 




pi 


o 


tP 


CO 


crs 


GN? 






C^ 


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Q5 


CO 




Ph 






, *" 1 




CO 



9 1 



GO 
CO 

CO 


CO 
OS 


CO 





CO -<* 
CO to 
cm tn 



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4) 



£ » 



ATPENDIX II. 23" 



APPEH DIX II. 

ASTRONOMICAL INSTRUMENTS. 

Astronomical Clock and Chronometer. 

1. — The order and succession of celestial phenomena make time a 
most important element in astronomy, and accordingly the utmost scientific 
and mechanical skill has been devoted to the perfection of instruments 
to indicate aud measure its lapse ; § 37. 

The best time-keepers now in use are the Clock and Chronometer. Both 
consist essentially of a motor, a combination of wheel-work to transmit and 
qualify the motion it impresses, and a check, alternately to arrest and 
liberate the movement, and thus to mark an interval designed to be some 
aliquot pai't of a day, the natural unit of duration. 

2. — The Clock. --In the clock, the motor is a weight A suspended 
from a cord wound about the drum B of a wheel C, and the check is 
the anchor escapement iV, controlled by the vibrations of a pendulum P, 
whose rod is geared to an arm projecting from the axis 0, with which the 
anchor is firmly connected. The weight A turns the drum B and its 
wheel C\ the wheel C turns the pinion I) and its wheel E\ the latter 
turns the pinion F and its wheel G, and so on to the pinion L and its 
wheel M, called the scape^wheel, of which the teeth are considerably under- 
cut, so as to turn their points in the direction of the motion. The flukes 
of the anchor are turned inward, forming two projections called pallets. 
The distance between the ends of the pallets is less than that between the 
points of two teeth that lie nearest the line drawn from one pallet to the 
other ; and no two teeth can, therefore, pass the same pallet without the 
wheel being arrested by the contact of a tooth on the opposite side with 
the other. 

With the swing of the pendulum the anchor oscillates, and one pallet 
is thus made to approach while the other recedes from the wheel. As soon 
as the receding pallet disengages itself from a tooth, the wheel is turned 



23$ 



SPHERICAL ASTRONOMY 
Fig. 6. 




by the motor and intermediate machinery till arrested by the approaching 
pallet, now interposed between its teeth on the opposite side. The re- 
turning swing of the pendulum reverses the pallet motion, liberates the 
wheel long enough for another tooth to pass, and again arrests it, and 
so on. 

Thus, by regulating the length of the pendulum and number of teeth 
on the scape-wheel, an index or hand connected with the arbor of the 
latter may be made to travel by successive leaps, as it were, around the 
circumference of a circle on the dial-plate in any given time. 

3. — If the anchor be connected with the seconds pendulum, and 
there be sixty teeth on the wheel, each leap will mark a second. The 



APPENDIX II. 239 

motions of the minute and hour hands are regulated by suitably propor- 
tioning the relative dimensions of the intermediate wheels with whose 
arbors these hands are connected. 

4. — The scape-wheel being in a state of constant tension by the 
incessant action of the motor, its teeth must act upon the pallets fiist by 
a blow and then by a pressure during the time of contact. The bearing 
surfaces, of which there are two on each pallet, inclined to one another, 
are so cut that the direction of the blow on the first from the tooth of 
the scape-wheel passes through the axis of the pendulum's motion, while 
the pressure from the same tooth on the second passes clear of that axis 
and accelerates the motion in the direction of the swing, thus restoring 
whatever of loss may come from friction and atmospheric resistance. 

5. — The pendulum bob possesses the principle of compensation. It 
consists of a cylindrical glass vessel resting upon a plate at the end of the 
pendulum rod. This vessel is filled with mercury to a depth so adjusted 
to the length of the rod as to elevate by its expansion or depress by its 
contraction the centre of oscillation just as much as this centre is depressed 
by the expansion or elevated by the contraction of the rod during a 
change of temperature. The distance between the axes of suspension and 
of oscillation being thus made invariable, the time of vibration will con- 
tinue constant, and the check be interposed at equal intervals. 

6. — Chronometer. — The chronometer is an accurately constructed 
balance watch, uniting great portability with extreme accuracy. It is of 
various sizes, the larger having dial-plates from three to four inches in 
diameter, and running from two to eight days between the windings. 
The larger kind are suspended upon gimbals to secure uniformity of 
position, are mounted in boxes, and are called box chronometers. The 
smaller kind resemble in shape and size a common watch, are worn in the 
pocket, and are called pocket chronometers. 

7. — The motor is an elastic spiral spring inclosed in a short cylin- 
drical box A, called the barrel, one end being permanently fastened to a 
stationary axis E, about which the barrel freely turns, and the other to 
the inner surface of the barrel. 

The barrel being turned in the direction of the coils of the spring, the 
elastic force of the latter is brought more and more into play, and its 
variable action thus produced is communicated by means of a chain B tc 
a variable lever C, called & fusee, whose office is to modify and transmit it 
uniformly to the works of the instrument. 

The fusee is a conical solid having its surface broken into a spiral 
shoulder, running from one end to the other, the curve being so regulated 



240 SPHERICAL ASTRONOMY 

that the distance of any one of its points from the axis of the fusee's mo- 
tion multiplied into the force of the spring, acting through the intermedium 
of the chain, shall be a constant quantity ; and as the main wheel D, which 




gives motion to the rest, is firmly secured to the fusee, the motion is made 
to act uniformly upon the instrument. 

8. — The swings of the pendulum by which the check was alternately 
interposed between and withdrawn from the teeth of the scape-wheel in 
the clock, are, in the chronometer, replaced by the vibrations of what is 
called the balance. This consists of a wheel B, freely movable about an 
axis (7, and a thin spiral spring S, one end of which is securely fastened 
to the hub of the wheel, and the other to a fixed support A. If when the 
spring is free from tension, the wheel 
be brought to rest it will remain so, 
just as a pendulum bob brought to 
rest at its lowest point will remain im- 
movable. If from this position of the 
wheel it be turned in either direction 
about its axis, the spring will wind or 
unwind, the elastic force of the spring will be called into play, and will, 
when the wheel is unobstructed, carry it back to its position of equilibrium. 
But having reached this position, its living force carries it beyond ; the 
action of the spring is reversed, and, after destroying the living force, 
will reverse the motion; the wheel will return to its position of equilibrium, 
which it will reach with a living force equal to that it had before at the 
same place, but in a contrary direction. The wheel will pass on, the action 
of the spring be reversed, the wheel will return as before, and thus the 
vibrations be continued forever, as in the case of the pendulum, but for 
the waste of living force from friction, atmospheric resistance, and absence 
of perfect elasticity in the spring. 

.9. — The angular acceleration impressed upon the balance by the spring 




APPENL1X II. 241 

is measured by the moment of its elastic force divided by the moment of 
inertia of the entire balance. When the temperature is increased, the 
spring is lengthened and the elastic force it exerts lessened ; the wheel is 
expanded, its matter thrown further from the axis of motiou, and the mo- 
ment of inertia consequently increased. On both accounts the angular 
acceleration is diminished, and the balance will vibrate slower, and the 
intervals between the checks be increased. The effect is just reversed when 
tlie temperature is diminished. This is the source of greatest difficulty 
with all portable time-keepers, and renders the common watch worthless 
for any thing beyond an approximate indicator of the time. 

10. — To remedy this defect, the common wheel is replaced by what is 
called an expansion balance, which is re- *iL^' 
presented in the figure. A A is a bar 
which receives the end of the arbor into 
an aperture at its middle point. To the 
ends of the bar are securely attached two 
compound metallic curves C C, composed 
of two concentric strips, one of steel and 
the other of brass, the latter being on the 
convex side ; these are soldered or burned 
together throughout their entire length. 
Each of these curved pieces carries a heavy mass D D, movable from one 
end to the other, but capable of being secured in any one place by means 
of a small clamp-screw shown in the drawing. 

Now when the temperature increases, the exterior brass expanding more 
than the interior steel, the ends C C are thrown inward towards the arbor, 
while the ends of the bar are thrown outward, but through a much less 
distance ; and thus by properly adjusting the places of the masses D D, the 
moment of inertia of the balance may be made to vary directly as the 
moment of the elastic force of the spring ; in which case the angular ac- 
celeration becomes constant, and the intervals between the interposition of 
the checks equal. 

11. — To regulate the rate, two large-headed screws B B, called mean- 
time screws, are inserted, one into each end of the bar. If the chronometer 
run too slow, the moment of inertia is too great for that of the spring, and 
these screws must be screwed up, which has the effect to lessen the dis- 
tance of their heads from the axis of motion, and thus to lessen the mo- 
ment of inertia, and increase the angular acceleration. If the chronometer 
run too fast, the screws must be unscrewed, the effect of which must be 
obvious. 

16 




212 SPHERICAL ASTRONOMY. 

12. — The escapement is of the kind usually called the deft eked, fn>m 
the fact that except at certain instants of time, the whole appendage of 
the balance-spring is relieved from the action of the scape-wheel. 

.<'--- C> v Fig. 10. 



The scape-wheel is represented at M\ it is urged by the motor, acting 
through the wheel-work, to move in the direction of the arrow-head. A 
steel roller C, called the main pallet, is firmly fixed to the arbor of the bal- 
ance. In the pallet is a notch t, having one of its faces considerably un- 
dercut, and covered with an agate or ruby plate to receive the action of 
the teeth of the scape-wheel. Securely fixed to one of the frame-plates of 
the chronometer is a stud B, and to this is attached a spring A, called the 
detent ; this spring is extremely thin and weak at the stud B. Attached 
to the detent is a stud 1). A ruby pin projects from the detent at c, which 
receives a tooth of the scape-wheel when one escapes from the pallet bear- 
ing i. From the stud D proceeds a very delicate spring E, called the lifting 
spring, which rests upon and extends beyond a projection F from the end 
of the detent; this projection being so made that the lifting spring cannot 
move in the direction from the scape-wheel without taking the detent with 
it, and thus lifting, as it were, the pin c from the tooth with 'which it is 
in contact, while it leaves the lifting spring free to move towards the scape- 
wheel without disturbing the detent. Concentric with the main pallet, at- 
tached to and just above it, is a small projecting stud a, called the lifting 
pallet, which is flattened on the face turned from the scape-wheel and 
rounded on the other. The flattened is called the lifting face. 

13. — Mode of Action. — In the position of the figure, the main pallet, 
under the action of the balance-spring, is moving in the direction of the 
arrow-head /, and the lifting pallet is coming with its lifting face in contact 
with the lifting spring E, which it lifts with the detent so as to raise the 
pin c clear of the tooth of the scape-wheel with which* it is in contact. 



1/* 



APPENDIX II. 2i3 

By the time tlte wheel is free from the pin c, the main pallet has advanced 
far enough to receive an impulse from the tooth t upon its jewelled surface 
?', and before this tooth escapes, the lifting pallet a parts with the lifting 
spring E, and the detent returns to its place of rest and interposes the pin 
c to receive the tooth i, as soon as the tooth t has been liberated by the 
onward movement of the main pallet from its face i. The balance having 
performed a vibration ty the impulse given to the main pallet, returns by 
the action of the balance-spring, and with it the lifting pallet a, whose 
rounded face, pressing against the lifting spring E, raises it and passes, first 
the detent without disturbing the latter, then the lifting spring, and moves 
on till the balance has completed the vibration, when it returns to the po- 
sition indicated in the figure, and the same evolution is performed again ; 
the balance thus making two vibrations for every impulse. 

The Vernier. 

1. — This is a device by which the value of any portion of the lineal 
distance between two divisions of a graduated scale of equal parts may be 
found in terms of the space itself. 

It consists of a scale whose length is equal to any assumed number oj 
parts of that to be subdivided, and is divided into equal parts of which the 
number is one greater or one less than the number of the primary scale 
taken for the length of the vernier. 

A B 223456 7 8 9 x o r 

-g-r~r~i , ! j i, i, i , i ,i , \ E 



i i r~> — r"T— i i i i - t~~r-i — rn II it i if 

C j c d r J> 

Let A D be any scale of equal parts, and denote by s the length of n— 1 
of these parts ; then will 

s 



»— 1 



be the value of the unit of the scale. Take a vernier BE of equal length 
s, and suppose it divided into n equal parts, then will 



be the length of one of its parts, and the difference of length between m 
parts of the scale and an equal number of parts of the vernier, will be 

ins ms m . s 

(i) 



tt— 1 n n(n—\)' 



I 



244 SPHERICAL ASTRONOMY. 

But is the value of the unit of the scale, and n the whole number of 

n— 1 

divisions of the vernier ; denoting the first by V, this difference may be 

written 

- V. 

n 

Now, the length of a part on the scale is greater than that on the vernier, 
and the number of parts on the vernier is greater by one than the number 
in an equal length on the scale ; hence, if the m th intermediate division of 
the vernier coincide with any one division on the scale, the zero of the ver- 
nier will fall between two divisions of the scale, and be in advance of that 
bearing the smaller figure by the distance expressed above ; so that, taking 
the zero of the vernier as the index or pointer, its distance from the zero of 
the scale will be the number of units denoted by the figure on the division 

next preceding, plus the — th part of the unit of the scale. Thus, in the 

figure, A being the zero of the scale, B that of the vernier and therefore 

the pointer, the distance of the latter from the former will be Aa + aB\ 

and because ?&=10, and the division b of the scale coincides with the 4th 

4 

of the vernier, w=4, and the distance AB=.Aa-\ . fe. 

' '' 10 J 

2. — The least value that may be read with certainty is obtained by 

making m=l, which will give, 

V 



Whence we have this rule for finding the lowest reading by means of the 
vernier, viz. : 

Divide the lowest count, or unit of the scale, by the number of divisions 
on the vernier. 

If the scale be tenths of inches, and we make ft =10, then will 

V 1 . 1 

n ~ 10 ' ~100' 

in which case the subdivisions will be carried to hundredths of inches, 

3. — The vernier is equally applicable to all kinds of scales, to circular 
as well as rectilinear: the only condition being that the different parts 
shall be equal. 

Suppose each degree on the circumference of a circle is divided into 6 
e^ual parts, and that the number of parts on the vernier is 60, then will 



APPENDIX II. 
F=10' = 800" 



245 



and 



V 600' 
n 



60 



10". 



So that the read: ng of angles with an instrument having such a circle may 
be carried to ten seconds. 

Micrometer. 

1. — The Micrometer is an instrument employed to make minute 
measurements, and is applicable alike to time and linear distance. It has 
various forms. 

1*. — The Reticle. — He who views a distant object through a telescope, 
does not look at the object but at its image within the tube of the instru- 
ment. The image of a point is always in a plane through the focus of the 
lens conjugate to the point itself, and perpendicular to the tube of the tel- 
escope. The visible portion of this plane is called the field of view. Some 
point in the field of view is arbitrarily assumed as an origin of reference, and 
marked by the intersection of a pair of cross wires. The line through this 
point and the optical centre of the field lens, is called the line of collimation. 

2. — If the telescope be at rest and an object in motion, the image of 
any one of its points will when visible pass across the field of view ; and 
one of the opaque wires being made to coincide with its path, the image 
will move directly towards the line of collimation, and the exact instant of 
its reaching it may be noted. But every such observation is liable to error. 
To increase the chances of avoiding this error, the wires marking the line 
of collimation are made perpendicular to one another, and an equal number 
of equidistant and parallel wires added on either side of that which is per- 
pendicular to the path of the image. When the motion of the image is 
uniform, an average of the times of passing the parallel wires will, accord- 
ing to the doctriue of chances, give a time of passing the line of collima- 
tion more free from error than the single observation. 

3. — This simple form 
of the micrometer is call- 
ed a reticle. The wires or 
spider lines are stretched 
across a circular metallic 
diaphragm pierced by a 
large concentric opening. On the edge of the diaphragm, and in the 
prolongation of the single wire, two studs project at right angles to its 
plane ; and these, with two antagonistic screws AB, hold the reticle in po- 





2<G SPHERICAL ASTRONOMY. 

sition ; the screws, for this purpose, passing through the tube of the teles 
cope and leaving the heads exposed for purposes of adjustment. 

4. — Position Filar Micrometer. — The pmpose of this instrument is 
to measure the angles at the observer, subtended by the distances between 
objects that appear very close together, and to determine the positions of 
the planes of these angles. It consists of two parts, viz. : One to measure 
the angle between the objects ; and the other, the inclination of the plane 
of the objects and observer to some co-ordinate plane. 

5. — The fii st is represented in the figure, a and c are two fine par- 
allel wires, which are made to move at right angLs to their lengths by 
means of screws fhmly connected with the foiks A and C, to whose prongs 
they are attached. The screws have fifty threads to the inch, and are 

Fig. 12. 



moved by nuts so mounted as to admit of a motion of rotation without 
translation, so that by turning the nuts a motion of translation is commu- 
nicated to the wires in either direction, depending upon the direction of 
the rotation. The outer surfaces of the nuts are cylindrical, and enter fric- 
tion tight the central perforations of two circular wheels whose planes are 
perpendicular to the lengths of the screws, and which are large enough to 
admit of their circumferences being divided into 100 equal parts, which 
parts are marked and numbered. Each wheel is provided with a station- 
ary pointer or index. 

A third and stationary wire, perpendicular to the first two, is supported 
by a diaphragm disconnected from the foiks. Upon one of the interior 
edges of this diaphragm, and parallel to its wire, is a graduated scale in 
the shape of a comb, having 50 teeth to the inch, so that one revolution of 
a nut will carry its movable wire f. om the centre of one valley between the 
teeth to that of the next. Near the central valley of the scale is a small 
hole to mark the zero of the comb-scale, from which the scale is estimated 
in either direction. It is easily seen that a turn of the nut-head through one 
of its divisions will move its wire through a linear distance equal to y-jg- of 
/ ff or 5o l o o °f an ^ ncn '■> aiJ d having ascertained by the measurement of 
some smai! distance on the ciivumfeience of a great ri e.'e of th<' celestial 
sphere, or by the process in Example, p. 249, its equivalent in arc, this, the 



APPENDIX II. 



247 




micrometer part of the arrangement, is readily applied to the determination 
of small angles. 

6. — The second and position part consists of a circular plate A A, 
called the position circle, some three or four inches in diameter, having 
its circumference divided into 360°, which are again subdivided to any 
convenient extent. The central part is cut away, and the micrometer 
arrangement so attached, with its wi es parallel to the position circle, as 
to admit of a free motion Fig. is. 

of rotation about an axis <# B= ^E 

through its centre, and per- 
pendicular to the plane of 
the wires. To the revolving 
plate of the micrometer part 
are attached two verniers 
V V, and motion is com- 
municated to the latter by a 
ratchet and pinion, of which 

latter the head is seen at O. The microscope by which the wires and 
comb-scale are magnified, and which serves also for the eye-glass of 
the telescope, is represented at E. By means of a screw cut upon a 
projecting ring ground the large and central aperture of the position circle, 
the instrument, as represented in the figure, is attached to the tail end of 
the telescope. 

7. — To measure the angular distance between two objects in the 
field of view, turn the head till the fixed wire passes through their 
images, then bisect the images by the movable wires ; note the reading 
on the comb-scale and upon the heads ; take their sum or difference 
According as the wires are on opposite sides, or same side of the zero of 
flie comb-scale. This reduced to arc will be the measure sought. Note 
also the reading of the position circle; this will give the inclination of the 
plane of the angle to the plane through the zero of the position circle. A 
second angle being measured in the same way, the difference between the 
second and first reading of the position circle will give the inclination of 
the planes of the two angles. 

Micrometer Revolution . 

The micrometer being supposed ir place, and the eye-piece pressed for- 
ward far enough to obtain a distinct view of the wires, the telescope is 
directed to some distant object, and adjusted to distinct vision. An image 
of the object will be formed on the plane of the wires, and any one of ..s 



24S SPHERICAL ASTRONOMY. 

linear dimensions may be measured by turning the position circle till the 
stationary wire coincides with, and the movable wires pass through the 
extremities of its image. The number of entire comb-teeth between the 
movable wires, multiplied by 100, and this product increased by the sum 
of the readings of the screw-heads, will give the linear dimensions of the 
image expressed in units of the screw-head. The value of the latter is, in 
the case we have taken, §- qVo °f an mc ^- To & n & the angle subtended 
by the object, we must know the angular value of the unit on the screw- 
head. 

It is demonstrated ( Optics, § 60) that the optical image of any point of 
an object, is on a right line drawn through the point and the optical cen- 
tre of the lens by which the image is formed. The angles, at the optical 
centre, subtended by an object and its image, are therefore equal, and if 
the images of objects which subtend equal angles were at the same dis- 
tance from the optical centre, they would be of the same size. The lineal 
dimensions of the images at the same distance from the optical centre, 
would therefore be proportional to the angles subtended by their respective 
objects, and to find the angular value in question, it would be sufficient 
to cause the image of some well-defined object, whose distance and dimen- 
sions are known, to be embraced by the wires, and to divide the angle which 
the object subtends, expressed in seconds, as determined trigonometrically, 
by the number of units of the screw-heads, which indicate their separation. 
But the distances and therefore the dimensions of images, whose objects 
subtend the same angle, are variable, being dependent on the distance of 
the objects, and from the value found by the above process must be de- 
duced that which would have resulted had the image been formed at some 
constant distance, which is that of the principal focus. 

Let / and /" denote the distances respectively of the object and its 
image from the optical centre, and F u the principal focal distance of the 
object-glass, supposed convex. Then, Optics, § 44, Eq. (40), 

Li - -f—L' 
f'~ f ' 

and denoting by n and N, the number of units of the screw-heads when 
the image is embraced at the distances f aud F lt respectively, we shall 
have, Optics, § 64, Eq. (58), 

/" : F u ::n: lY; 



whenc3 



F f—F 



APPENDIX II. 249 

and calling a, the number of seconds in the angle subtended by the object, 
we have, by the rule just given. 

a a ./ 



JST n.(f-F„) 



(a) 



Example. — The length of the object measured in a direction perpen- 
dicular to the line of sight was 3 feet ; the distance from the object-glass, 
261.9 yards; the principal focal length, 45.75 inches; and the sum of the 
divisions on the screw-heads indicating the separation of the wires, 1819. 
Then 

/= 261.9 yds -; F u = 45.75 in = 1.2708 yds -; n= 1819. 
f—F n = 260.6292 yd \ 



whence 



Log 


. a 


. 




u 


f 


• 


• 


.1 


n 


a 


comp 


u 


f- 


-F u 


u 


a 


a 


= 0' 


'.4351 



R 5 yds- 
tan I- a = , , of which the W. is 7.280835 ; 

2 261.9 yds -' ° ' 

a= 13' 07 // .57 = 787".57. 

2.8962892 

2.4181355 

. —4.7401673 

. —3.5839923 

. —16385843 



Now, to measure the angle subtended by the distance between any two 
points, direct the telescope so as to get the images of the points in the 
field, and turn the micrometer till the stationary wire apparently passes 
through them, and by a motion of the scrow-heads bring the movable 
wires to the images — the number of units of -he screw-head, which indi- 
cate the separation of the wires, multiplied by the decimal 0".4351, will 
give the number of seconds in the angle. 

The value of -==., being a function of F u , Eq. (a), will .of course vary 

with the object-glass, but is perfectly independent of the eye-glass. 

If the distance/ be so great that F u may be neglected in comparison, 
then will Eq. (a) give 

N — n, 

which will be the case when the ang'ilar value is determined from astro- 
nomical objects. 



250 



SPHERICAL ASTRONOMY. 



Spirit-Level. 

1. — This is an instrument used to adjust a line to a given position in 
reference to the horizon. 

It Consists of a cylindrical glass tube A A, whose axis is the arc of a 
circle. This tube is rilled nearly full with some one of the more perfect 
fluids, such as alcohol or naphthalic ether, leaving a small portion of air, 
seen at i>, called the air-bubble, and hermetically sealed at both en Is. It 




is then usually set in a metallic tube (7, very much cut away on one side 
from the middle towards the ends, so as to exhibit the bubble and fluid 
when in a horizontal position. This metallic tube is connected with a 
plate of metal F F. by a hinge E and screw D, the axis of the hinge 
being perpendicular, and that of the screw parallel to the plane of the 
circular axis of the level. 

2. — A scale of equal parts is cut either upon the upper surface of the 
glass tube or upon a slip of ivory and metal lying in the plane of the tube's 
curve, as represented at G G. The divisions of the scale being numbered, 
the value of the spaces in arc is readily ascertained by attaching the level to 
the face of a vertical graduated circle, and turning the latter sufficiently to 
cause the air-bubble to pass from one end of the scale to the other. The 
angular space passed over by the circle reduced to seconds, divided by 
the number of units on the scale traversed by the bubble, will give the 
value of the unit in some multiple of the second. 

3. — Use, — The surface of the fluid being always horizontal, the line 
connecting the ends of the bubble will be a level chord of the level's arc, 
and the radius passing through the point of the scale midway between the 
ends of the bubble will be vertical. 

Now, suppose any line of an instrument with which the level is used 
to be made parallel either to the radius passing through the zero of the 
scale, or to the chord whose ends are marked by the same numbers ; 
then, to make this line vertical in the first case, or horizontal in the 
second, move the instrument, the level being securely attached, till the 
ends of the bubble are equally distant from the zero. 

If the ends of the bubble be not at the same distance from the zero, 
the inclinat'on x of the line in question to the vertical or horizontal 



APPENDIX II. 251 

direction is thus found : Let a denote the semi-length of the bubble, 
m and n the numbers of the scale at its extremities, then will 

a-\-x=m, 
a — x=n ; 
whence 

x =-^r= l ( 2 ) 

This value of x being independent of the length of the bubble, which 
is -indeed a variable quantity, even in the same level, because of its varying 
temperature, gives the inclination of the line under consideration to its 
proper position, when the level is adjusted to the instrument. 

If the lower surface of the plate F F be parallel to the chords of equal 
numbers, the inclination of any given line or plane may be ascertained by 
laying this plate upon it and applying the above rule. 

But if the lower surface of the plate be not parallel to the chords of 
equal numbers, its inclination to them, and that of the plane or line in 
question to the horizontal or vertical direction may nevertheless be found 
thus : Denoting the first by y, and the latter, as before, by .?, and using 
the notation of equation (2), we have for one position of the level, 

x=l-y, 

and for the reversed position of the plate with its level, 

x=l'+y, 
whence 



l-\-V m—n-\-m'—n' 
X = ~2~ = I ' 



I — V m — n—m' — n' 
~ 2 4 

If the given surface or line be provided with adjusting screws, as is 
the case in all astronomical instruments, the ends of the bubble may be 
brought to the same reading in the first position of the level, in which 
case, we have m=n, and 

m' — n' 

* = — T -=-y • • • ■ (3) 

The angle y is called the error of the level, and the angle x the error 
in level of the instrument, and the above equation gives this rule for 
finding and correcting these errors, viz. \ 



252 SPHERICAL ASTRONOMY. 

The level being placed over the given line, bring, by means of the 
adjusting screws of the instrument, the bubble to read the same at both 
ends ; then reverse the level, or turn it end for end, and take one-fourth 
of the difference of the new readings ; add this to the lesser of the read- 
ings, and turn the screw D till the end of the bubble nearest the zero 
reach the nurnbt r answering to this sum, to which add again the same 
quantity, and bring the end of the bubble to this new reading by the 
adjusting screws of the instrument. The ends of the bubble will stand at 
the same numbers, and both errors will be destroyed. 

Reading Microscope. 

1. — This instrument, like the vernier, has for its object to read and 
subdivide the space between two consecutive divisions of any scale of 
equal parts, and is the most perfect yet devised for this purpose. 

It is a compound microscope, whose object-glass forms an enlarged 
image of the space to be divided. This image is thrown upon the plane 
of two spider-lines or wires, arranged in the form of a St. Andrew's cross, 
and so placed that a line bisecting its smaller angles is parallel to the 
cuts or division marks of the scale. The cross is attached to a diaphragm, 
which is moved by a micrometer screw in the direction of its plane, per- 
pendicular to the axis of the microscope. The head of the screw is 
divided into any number of equal parts, depending upon the nature of 
the scale and the extent to which the subdivisions are to be carried. The 
numbers on the head are so placed that when the screw is turned in the 
direction to bring them in the order of their increase to a fixed pointer, 
the cross shall move along the image-scale in the direction in which its 
numbers decrease. 

Within the barrel of the microscope is a stationary comb-scale, like that 
in the position micrometer. Its plane is parallel to that of the cross, and 
the distance between the centres of two valleys, separated by a single tooth, 
is equal to the space over which the cross is moved by a single revolution 
of the screw. Every fifth valley is cut deeper than the others to facilitate 
the reading ; and near the bottom of the central valley of the comb is 
a small circular aperture, to mark the zero position of the pointer or 
index, which is a small wire attached to the movable diaphragm, and so 
placed that its prolongation shall bisect the smaller angles of the cross. 

In (1), A A is the main tube of the microscope, passing through a collar 
or support B, where it is firmly held by two milled nuts g g, which act 
upon a screw cut upon the outer surface of the tube. These nuts also 
serve to change *he distance of the whole microscope from the scale to 



APPENDIX II. 



253 



be read ; h is the object-glass placed in a smaller tube, upon whose outer 
surface is also a screw, by which this glass may be moved independently 



Fig. 15. 




Fig. 16. 2 




of the main tube ; the diaphragm of the cross is in a working box, wnose 
edge is seen at a. ; e is the graduated head, firmly attached by a friction 
clamp to the nut b of the micrometer screw ; f is a pointer attached to 
the working box ; d is the eye-glass, which moves freely in the direction 
of the axis of the microscope by a sliding tube ; at c' is represented the 
head of a small screw, which supports and gives motion to the comb-scale 
within the working box, and S S represents the edge of the scale to be 
subdivided. In (2) is represented the field of view, as seen when the eye 
is applied at c?, in which m m' is the image of the scale, with one of its 
cuts bisecting the smaller angles of the cross, and e the wire index at 
its zero position, as indicated by its being seen through the centre of the 
circular aperture of the comb. In this position of the pointer, the zero 
of the graduated head e is brought to the index /, by holding the nut b 
firmly in the hand, and turning the head, which is only held in its place, 
as before stated, by the action of the friction nut. 

2. — The quotient arising from dividing the length of the image space 
by that over which the wires move in one revolution of the screw-head, 
as given by the comb-scale and head, is called the run of the micrometer. 
For convenience, the run should be an entire number. 

3. — The image-scale must be accurately in the plane of the wires, 
otherwise there would be a parallactic motion, which would shift the 
position of the wires on the image-scale at every change in the position 
of the eye, and thus vitiate the measurement. This parallactic motion 
is easily detected by slightly shifting the position of the eye when looking 
through the eye-glass. 

There are, then, two adjustments for the reading microscope, viz., that 
for the run and that for parallax 



254 SPHERICAL ASTRONOMY. 

4. — The size of the image of an object, and its distance from the 
lens by which it is formed, are dependent upon the distance of the object 
from the lens, being greater in proportion as this distance is less, and less 
hs it is greater. 

If the distance of the object-glass of the microscope from the scale 
be changed by means of the screw on the tube at A, the size of the image 
space will be altered, and may, therefore, be made of such dimensions 
that the cross w T ill move from one division to the next in order, by a given 
entire number of revolutions ; and if by this operation, the image be 
thrown off the plane of the wires, as it in general will, it is restored by 
changing the distance of the whole body of the microscope from the 
scale by means of the milled nuts g g. By two or three efforts cautiously 
conducted, the adjustments may be made without difficulty. 

To illustrate, let the scale be that of the sexagesimal division of the 
circle, and suppose each degree divided into twelve equal parts, each 
space will be equal to five minutes ; if we make the run five, each tooth 
on the comb will be equal to one minute, and if the screw r -head be divided 
into sixty equal parts, each of its spaces will be equal to one second; so 
that the circle may be read to seconds. 

Now suppose on examining the run, which is done by turning the 
screw-head till the cross moves from one, division to the next in order, it 
be found h r 10" ; it is too great. Move the object-glass k from the plane 
of the circle by screwing in its tube, the image will decrease, and, if it 
were before on the plane of the wires, it will now pass to some position 
between that plane and the object-glass h. Move the whole body of the 
microscope by means of the milled nuts gg towards the circle ; the image 
will be restored to its proper position, with less dimensions than it had 
before. By one or two repetitions of this process the adjustments are 
made. 

o. — The wire pointer at its zero position on the comb-scale is the 
index of the circle or instrument scale. When the pointer, in this po- 
sition, is immediately opposite a division mark of the circle scale, say the 
third after that marked 27°, which is indicated by the angles of the cross 
being bisected by the image of that division mark, the reading is 
27° 15' 00" ; but if the intersection of the cross wires falls between the 
third and fourth divisions after that marked 27°, then will the reading be 
greater than that above by the value of the distance from the cross wires 
to the division mark to which the cross will move by turning the screw- 
head in the order of its increasing numbers. To find thin value, turn de 
screw-head in the direction just indicated till the angles cf the cr^s? ars 



APPENDIX II. 



255 



bisected by the division mark in question, and count the entire number 
of comb teeth between the aperture and pointer, then note the reading 
on the screw-head ; suppose the former to be 3 and the latter 41, the 
true reading will be 27° 18' 41''. 



The Transit 
1- — The transit is an instrument which is used in connection with 



ime-piece to ascertain the precise instant of a body's passino- tl 



a time 



le me- 




256 SPHERICAL ASTRONOMY. 

ridian of a place. It consists of a telescope T T, usually of considerable 
power, permanently fixed to a substantial axis A A, at light angles to its 
length. The axis terminates at each end in a steel pivot, accurately 
turned with a diamond point, to a cylindrical shape. The pivots are of 
equal diameters, received into notches cut in two blocks of metal, called 
Ys, which rest in metallic boxes, the latter being imbedded in metallic or 
stone piers, according as the instrument is intended to be portable or fixed. 

2. — Permanently attached to the tail or eye end of the telescope, 
on opposite sides, are two small graduated circles, called finders. The 
planes of these circles are perpendicular to the axis of the transit, and each 
circle has an index-arm, which carries a small spirit-level and two verniers, 
one at each end. The index-arms are movable about the centres of their 
respective circles, and are, as well as the axis of the transit, provided 
with a clamping and tangent screw arrangement, thus affording, with the 
aid of the level and verniers, the means of giving the telescope any de- 
sired inclination to the horizon. 

3. — At the solar focus of the object-glass of the telescope is a 
reticle, Fig. 11, in which the single is replaced by a double wire, with small 
interval, and so placed as to be parallel to the axis of the transit. These 
are called axis wires. Those wires of the reticle which are at right 
angles to these are called the normal wires. To the fixed wires of the 
reticle a movable one is added ; it is always parallel to the normal wires, 
indeed, is itself a normal wire, and is put in motion in the direction of 
the axis wires by means of a. micrometer screw, with graduated head, 
shown at m. 

4. — The small tube containing the eye-piece of the telescope is 
attached to a sliding-frame, connected with a screw e, by which the eye- 
piece is carried from one side of the field of view to the other, in the 
direction of the axial wires. 

5. — The axis is hollow throughout, and the pivots are perforated 
at the ends to admit the light from a lamp L, supported upon one of the 
piers. This light is received by a reflector within the tube of the tel- 
escope, and inclined to its axis under an angle of 45°, and is reflected to 
the eye-glass, thus illuminating the field of view, and exhibiting the wires 
of the reticle. The reflector is perforated by an elliptical opening in its 
centre, to permit the direct light from any external object to pass freely 
to the eye end of the telescope. When the illumination is through the 
other end of the axis, the reflector is revolved through an angle of 90°, 
by means of a milled-headed wire, with which it is permanently con- 
nected. The head is shown at r. 



APPENDIX II. 



257 



Fig. 18. 





Fig. 20. 



6. — The boxes which support the Ys are large enough to permit 
i slight play in the latter; one in a horizontal, Fig. 18, and the other in 
a vertical direction, Fig. 19, the motions being effected by antagonistic 
screws. By the first of these motions, 
the line of collimation is brought 
to the meridian, after the rougher ap- 
proximations to that plane are made 
by other means, and by the second 
the axis is made horizontal by the 
aid of a large and delicate spirit- 
level, Fig. 20, mounted upon in- 
verted Ys, far enough apart to rest 
upon the pivots. 




Adjustments. 

7- — The transit is adjusted within itself when its line of collimation 
is perpendicular to its axis ; and it is in position, when its axis is perpen- 
dicular to the meridian. Its finders are adjusted, if the air-bubbles at 
their levels indicate the same reading at both ends, when the verniers 
indicate the true inclination of the line of collimation to the vertical or 
horizon. 

8. — It is by no means necessary, or even desirable, to aim at 
perfect adjustment. It will, in general, be much safer to reduce the 
errors of adjustment to narrow limits, then to determine their amount, 
and eliminate their effect from observation, in the manner to be described 
presently. 

9. — Line of Collimation. — Direct the telescope to some small, 
distant, and well-defined terrestrial object. Bring it apparently between 
the horizontal wires, and measure its distance from the central norma] 
wire by means of the micrometer and movable wire ; denote this dis- 

n 



25S SPHERICAL ASTRONOMY. 

tance by c'. Lift the transit from its Ys, turn the axis end for end, and 
measure, as before, the apparent distance of the same object from the 
middle wire, and denote this distance by c" . Place the movable wire at 
the distance, of 

c'+c" 



on the side of the object from the middle wire, and move the whole 
reticle by the antagonistic adjusting screws, which lie in the direction of 
the axial wires, till the object appears on the movable wire ; the line of 
collimation will be adjusted. 

10. — Error of this adjustment. — If n denote the value in arc of 
the micrometer's unit, then will the angle which the line of collimation 
makes with its proper position, before moving the diaphragm, be 



(4) 



and the line of collimation will describe, when the telescope is moved, 
a conical surface, whose intersection with the celestial sphere will be a 
small circle. 

Example. — When the telescope is pointing to the south, let the middle 
^wire appear to be 326.3 revolutions to the right hand of the object ; 
when the axis is reversed, let it appear 318.7 to the right, then will 

326.3 — 318.7 

n . =zc=3.8n, 

2 

and if one revolution of the micrometer correspond to the space an equa- 
torial star would pass over in three seconds of time, then will 

35.x 15 ... Mtm 

n = = 0".45, 

100 ' 

and 

c=3.8x0".45 = l".7l. 

11. — The axis. — This must first be levelled, then moved in azimuth, 
till it is perpendicular to the meridian. 

Mount the level with its inverted Ys upon the pivots, bring the bubble 
to the same reading at each end by the adjusting screw of the level ; 
reverse the level, and bring the bubble again to the same readings — half 
by the screw of the level and half by the vertical antagonist screws of 
the Y, which admits of vertical motion. Repeat the operation once or 
twice, and the thing is done. 



APPENDIX II. 259 

12. — The error in tJ is adjustment. — After the first approximation, 
denote by c', e", &c, the reading of the east end of the level; by w\ w", 
&c, the same of the west end, and let the parenthesis denote the end of 
the axis marked by some peculiarity, such as the clamp, or illumination ; 
then mounting the level in its place, and writing its readings in any one 
position upon the same horizontal line, we may have 

First position of level . . . . e' (w') 

Level reversed e" ( w ) 



it ^f f e '+ e ' 

Halt sums ot — — - 



These half sums are the readings which the level would have indicated in 
both positions had it been in perfect adjustment, and 



the error, or inclination of the axis to the horizon, expressed in the level's 
unit, provided its pivots be of the same size. But lest there may be a 
difference in the pivots, reverse the axis, and apply the level as before, 
and we may have 





For first position of level . . (e" ; ) . . . w" T 
Level reversed { e "") • • • w "" 




ILUf-um- («'")+(*"") . *"'+«"" 


when 


2 2 
ce 


and 


w"' + w""-(e"') + (e"")_, 
4 




s—s r {w') + {w")-\-{e"') + (e"")-w"' + w""+e'+e" 



10 



(6) 



(?) 



will be the angle which the axis makes with the line whose inclination is 
given in equation (5), whence, denoting the inclination of the axis to the 
horizon, or the angle which a plane perpendicular to the axis makes with 
a vertical plane at right angles to the projection of the axis, on the hori- 
zon, by V , we shall have 

This value is expressed in terras of the level's unit ; if »' denote the 



260 SPHERICAL ASTRONOMY; 

value of this unit in seconds, we shall have, representing tlje angle in 
seconds of arc by I, 

l=n' V=n'(s±t) '. . (8) 

The value of t for the same axis is constant, and must be determined 
by taking a mean of a great many careful observations. If it be positive, 
the pivot at the clamp end of the axis is the larger, but if negative, it h 
the smaller. 

When the half sum of the readings on the west end is greater than 
that on the east, the inclination is counted positive, and the plane perpen- 
dicular to the axis will fall to the east of the zenith ; and as it is obvious 
that the axis will be depressed on the side of the greater pivot, when the 
level indicates perfect adjustment, the upper sign, in equation (8), must 
be taken when that pivot is to the east, and lower when to the west. 

Example. — Performing the operations indicated, let the following be 
the record, viz. : 

First position of level . , 
Level reversed .... 



71.40 
78.60 


(87.60) 
(80.10) 


150.00 
167.70 


167.70 


4)17.70(4.425 = 


s 


(73.95) 
(81.30) 


84.90 

77.85 



' Axis reversed. 

First position of level . . 

Level reversed (81.30) 

155.25 162.75 
162.75 

4)7.50(1.875 = s' 

Adding the indications of the level diagonally, we have 

(822.95)-812.Y5 
16 

Applying the level to the face of some vertical graduated circle, § 95, 
let 23.5 of i*s units correspond to 30" then will 

23.5 

Whence for 

Clamp end west Z=(4.425-0.637)xl"-276 = 4".83348 
Clamp end east Z=(1.875 + 0.637)Xl .27G = 3".205312 



APPENDIX II. 



261 



13. — Azimuth adjustment. — It is now supposed that the errors of 
collimation and of level are destroyed. By a reference to a map of the 
stars it will be seen that a straight line drawn from the Pole star to a 
point midway between the fifth and sixth stars, called s and £ respectively, 
in the constellation of the Great Bear, will pass sensibly, over the pole. 
About the time when this line assumes a vertical position, direct the tel- 
escope to the Pole star, and keep its image on the middle normal wire 
by a motion of the horizontal adjusting screws of the Y, or by the mo- 
tion of the Ys themselves, if the requisite range be beyond that of the 
screws, and at the instant when it is inferred from a suspended plum- 
met, that the line referred to is exactly, vertical, arrest the motion and 
secure the Ys. The adjustment will be sufficient for the first approx- 
imation. 

Next find the amount of azimuth error. The axis being horizontal, and 
the line of collimation perpendicular to the axis, it is plain that in the 
motion of the telescope the line of collimation will describe the plane 
of a vertical circle, and that the angle made by this plane with the me- 
ridian is the error in question. 

Let H OR be the horizon, RZH 
the meridian, P the pole, Z the ze- 
nith, and S the star when on the 
line of collimation. Make, 

X =latitude of place=90° — ZP; 

S = declination of star =90'?— PS, 
positive when north, nega- 
tive when south ; 

P=Z P £=hour angle of star ; 

Z =HZS = azimuth of star's position, and equal to the error sought 
when east. 

2 =ZS= zenith distance of star. 




Then, in the triangle Z P S, 



sinP: 



sin Z . sin z 

cos 6 ' 



and because the sines of P and Z are very small, 



p= sin (\- 8) z 

cos S 



(») 



in which P and Z are expressed in seconds of arc 



262 SPHERICAL ASTRONOMY. 



Divide both members by 15 and make 
sin (X — S) 



cos S 
and equation (9) becomes 

P Jc .Z 



=*; 



(10) 



15 15 

P . 

in which — is the time required for the star to pass from the vertical de- 
15 

scribed by the line of collimation to the meridian, and if / denote the time 

indicated by a timepiece at the instant the star is on the central normal 

wire, the time of meridiin passage will be 

H-*4 = T (") 

Let e be the error of the timepiece at the time t referred to the vernal 
equinox ; m the rale or quantity by which this error is increased or di- 
minished in one day or twenty-four hours ; then, if R denote the right 
ascension of the star, supposed known, will 

t+e+Jc. — = R (12) 

and for a second star 

t'+e+(t'-t)m+Jc r . — = R\ 

in which t' — t is reduced to the decimal part of a day. Subtracting the 
first from the second, we get 

t'-t±(t'-t)m + (Jc'~k).^ z = R'-R, 
lo 

in which the upper sign is used when the timepiece runs too slow, and 
the lower when too fast ; whence, 



Z R'-R- (t'-t )±jt-t)m 

15= F=k .... (13) 

Z is hence known, and for which the instrument may be corrected, 
if desired. This value in equation (11), gives the time of meridian pass- 
age, and in equation (12), which may be written 

e= R-t-k. — = R-T, (13') 

gives the error of the timepiece. 



APPENDIX II. 



263 



The sign of the quantity k changes when /he declination of the star 
exceeds the latitude, and also when the star passes below the pole, since 
in this latter case S becomes 90°, plus the polar distance. The right 
ascension for all stars which pass below the pole must be diminished by 
twelve hours. 

Using the Polar star in its upper and lower passage instead of two 
separate stars, equation (13) becomes 



15 



12 h — (f — t)±(t' — t)m 
k'+k 



■ (14) 



Fig. 22. 



When three consecutive transits of the Pole star are observed, and the 
intervals are equal, Z will be zero, and the transit's axis is perpendicular 
to the meridian. 

The values of k and k\ in equation (13), must be found from stars 
differiug at least 50° in declination. 

14. — Let it now be supposed that after adjusting the transit in the 
manner explained, there is still (as in general there will be) remaining a 
small error in collimation, level, and azimuth. It remains to be shown 
how the effects of these may be eliminated from the observation, and a 
result obtained the same as though the instrument had been perfect. 

Let all the circles referred to in what 
precedes be projected on the horizon, 
represented in M E R Q. Let Z be the 
zenith ; P the pole ; M Z R the merid- 
ian ; VZ V a vertical circle at right 
angles to the projection of the axis of 
the transit ; Vs'V the circle at right 
angles to the axis ; Cs C the parallel 
small circle cut from the celestial sphere 
by the motion of the line of collima- 
tion ; E Q the equator; and esg the 
diurnal path of a star. 

When the star appears on the central normal wire, it will be at s ; and 
if the time be noted and increased by the angle s P 0, expressed in 
time, we shall have the indication of the timekeeper when the star is on 
the meridian. Now, 

sPO = sPs'+s'I?s"+s' r PO', 
the angle sPs' is measured by an arc of the equator, which is equal to 




264: SPHERICAL ASTRONOMY. 

i s' divided jy the cosine of the distance of s s r from the equator, which 
distance is the declination of the star. But 

c being as before the error of collimation ; hence, 

sPs' = 



cos 8' 



The angle s' Vs" is the error of the level denoted by I. Then re- 
garding Ps"V as the arc of a great circle, from which it will differ by 
an inappreciable quantity within the limits of the supposed errors, we 
shall have, in the triangle Ps'V, writing the small angles for their sines, 

*'P C "_7 SmFs '_; COSJX-^) 
SID Ps COS 

representing the zenith distance by (X — (5"), to which it is nearly equal, and 
regarding Vs' as the altitude, from which it differs but by a very small 
quantity. 

The angle s"P is given by equation (9), Z denoting as before the 
azimuth error. Whence, denoting by t the time of observation, we obtain 
for the time of meridian passage 

c I cos (\-8)Z sin(X-a) 

t+— J + TT- 1 r — • 3 — = 1 . . (15) 

15. cos o 15 cos o 15 cos 6 v ' 

in which c, I, and Z may be found in the manner already indicated, or 
still better as follows. 



Making 



1 a 



1 5 . cos S 
cos(X— 8) 



15 . cos S 

sin (X— S) _ z 
1 5 . cos 8 ~ 



=L \ ■ • • (16) 



and supposing the timepiece regulated by the vernal equinox, and rep- 
resenting its error at the time t by e, and denoting by R, the right ascen- 
sion of the star, we obtain 

t+e+c. C+l.L+Z .Z, = R . . . . . (11) 

in which, if e, c, I, and Z be regarded as unknown, their values may be 
found by carefully observing four stars, whose positions are well known, 
and which differ but little in right ascension, and considerably in declina- 



APPENDIX IT. 265 

tion The values of C, X, and Z x being computed in eacli case from 
equation (16), we may have 

t' +e'+c.C +I.L' +z.Z x ' =E f 
t" +e'+c.C" +I.L" +z.Z x " =R" 
t"> +e >+c . C" +1 . L"' +z . Zr =R"' 
+e'+c . C"" + l . L""+z . Z x ""=R n " 



ittti 



which are sufficient. But as there are always slight errors in the obser- 
vations themselves, it would be well, where great accuracy is required, to 
increase the number of these equations, and treat them after the method 
of least squares. 

15. — The finding circles. — These may indicate zenith distances, 
altitudes, or polar distances. The rule for adjusting is the same for all. 

Direct the telescope to the distant horizon, and move it till the image 
of some small object appear midway between the double axial wires : 
clamp the axis, move the index-arm till its level indicates the same read- 
ing at both ends of the bubble, and note the reading of the vernier. 
Unclamp and reverse the axis ; bring the image of the same object again 
between the same wires, and clamp the axis ; move the index-arm till the 
bubble has the same reading at each end, and again note the reading of the 
vernier. If the vernier reading be the same as before, the circles are in 
adjustment ; if not, add the readings together, take the half sum, move the 
index-arm till the vernier is brought to the reading indicated by this half 
sum, clamp the index-arm, and bring the air-bubble so as to have the same 
reading at each end by the adjusting screws of the level. It would be well 
to verify by repeating the process. It may be, that the finders are gradu- 
ated from 0° to 360°, in which case, if the first reading were a°, the 
second ought to be 360° — a°. 

16. — The adjustments in azimuth, colli mation, and level being per- 
fected, the middle normal wire will be a visible representation of that 
portion of the celestial meridian to which the telescope is pointed ; and 
when a star is seen to cross this wire in the telescope, it is in the act of 
culminating. The precise instant of this event being noted by the clock 
or chronometer, the time of meridian passage is known, and any error 
in noting this precise time is lessened by the use of the lateral wires of 
the reticle, as already explained. 

17. — Besides, these lateral wires increase the chances of securing an 
observation that might, without them, be lost. It frequently happens 
that efforts to obtain the time of a body's passiug the middle or other wire 
are defeated by the presence of clouds, or other accidental circumstances, 



266 



SPHERICAL ASTRONOMY. 



in which, if the time of passing any one be obtained, that of passing the 
middle or mean place- of the wires, when not equally distant, may be 
"educed thus. 

Let t u t 2 , £ 3 , &c, be the times of crossing the several wires in order, 
hen will 

h + U+t z +...t n j 



(18) 



n which t m denotes the time of the body's crossing the mean position of 
the wires, and n the number of wires. And 

{t m — *i).cos 3=*,, 

(t m —t 2 ) . cos 5=4, 

(t m —h).cosd=i 3 , y (19) 

(t m —t n ) . cos S=zi n . 

in which S denotes the declination of the body observed, and i x , i 2 , i 3 . . . i„ 
the constant intervals of time required for a body in the equator to pass 
over the distances which separate the several wires from their mean 
position. 

Adding equations (19) together, we obtain 



2t 2i 

t m = — + . 

n n cos 



(20) 



in which 2 denotes the algebraic sum of the quantities expressed by the 
letter written after it. 

By carefully observing a star whose declination is known, we obtain 
the values of i u i 2 , &c. ; and these being tabulated with their proper signs, 
equation (20) will give the time of a body's passing the mean position 
from the time of passing one or more of the threads. 

The Collimating Telescope. 

1. — In some situations it would not be possible to obtain a distant 
mark by which to collimate, and a near one could not be used in conse- 
quence of its image falling too far behind the reticle. In such cases 
recourse must be had to what is called the collimating telescope. 

Fig. 28. 




This is a telescope whose eye-piece is removed, and upon its tube is 
mounted a small swing-frame, supporting a reflector, by means of which 



APPENDIX II. 



267 



sufficient light may be thrown through the telescope to illuminate a pair 
of cross wires, situated at the solar focus of the object-glass. 

In this position of the wires, we have, from the principles of optics, 
these facts, viz. : the rays composing the pencil of light proceeding from 
any point of the cross, will emerge from the collimator parallel to a line 
drawn through that point and the optical centre of the lens ; and if the 
telescope of the transit be directed towards the collimator so as \o receive 
these rays, an image of the point in question will appear in its solar focus, 
and on a line drawn through the optical centre of its object-glass, par- 
allel to these same rays. 



Fig. 24. 



The Vertical Collimator. 

1. — This instrument is used for the double purpose of collimating, 
and for finding the zenith or horizontal point of circles, used in the meas- 
urement of vertical angles. It consists of a collimating telescope T 
mounted in a vertical position upon 
an annular plate JR, of cast-iron, float- 
ing upon the free surface of mercury, 
contained in an annular trough S, also 
of cast-iron. The annular plate is called 
the float. The telescope is mounted 
upon the float in a manner similar to 
the transit, except that the axis is near- 
er to the object end. One of the Ys 
may be elevated or depressed by an ad- 
justing screw A, while the telescope is 
turned about its axis by another A f , thus 
affording the means of giving the line 

"joining the cross wires and the optical centre of the lens a vertical position. 
L is the lamp, and G the reflector, to catch its light and throw it upon 
the cross wires at the lower end of the tube. 

2. — The collimating process. — Take the transit for instance. Level 
the axis carefully ; turn the telescope in a vertical position ; place the 
collimator below, and bring the image of the intersection of its cross wires, 
seen upon the bright ground G, accurately on the intersection of the 
middle wires in the transit, by means of the adjusting screws of the 
collimator; next turn the float in azimuth through 180°. If the emer- 
gent rays from the collimator be vertical, the image of the intersection 
of the collimator's wires will remain stationary, but if not, the image will 
move in the circumference of a circle ; because, the plane of floatation 




268 



SPHERICAL ASTRONOMY 






remaining the same, the emergent rays from the collimator will preserve 
their inclination to the horizon unchanged, thus causing the line through 
the optical centre of the transit's lens, and parallel to these rays, to de- 
scribe a conical surface. The axis of this cone, which is a vertical line, 
is the position for the line of collimation. Supposing, then, the image 
to have changed its position during the semi-rotation of the float, renew 
the contact of the image and wires ; one half by the adjusting screws of 
the collimator, and the other half by a motion of the transit and the 
adjusting screws of the diaphragm of its wires. This process being re- 
peated once or twice, the adjustment is made. 

3. — The zenith or horizontal points. — Direct the telescope of any 
circle to the collimator, and bring the image of the intersection of the 
cross wires in the collimator to the line of collimation ; read the circle, 
and revolve the float through an azimuth of 180° ; renew the contact of 
the image line of collimation by moving the circle, if necessary, and read 
again ; denote the first leading by a, the second by a', and that of the 
zenith point by z, and we have 

a-\-a' 

2 = 180°+——; 



and denoting the reading of the horizontal point by h, 



h= 



a+a' 



90°. 



Fig. 25. 



The Collimating Eye-piece. 

4. — If now the swing-frame and its reflector be transferred from the 
collimating telescope to the eye-piece of the telescope of the instrument sup- 
posed to be vertical over a basin of mercury, this latter telescope becomes 
its own vertical collimator by reflection, on applying the lamp to the swing 
reflector. By perforating the swing reflector, and applying the eye behind 
it, two sets of wires will be seen in the solar focus of the telescope, and 
the collimating process consists in making the wires of 
one of these sets coincident with those of the other, by 
the joint motion of the telescope and its reticle. The 
little swing reflector, with a single microscope as an eye- 
piece, just behind its perforation, to magnify the wires 
and their images, constitutes the collimating eye-piece. 
This beautiful little instrument, which has done so much 
to facilitate the process of collimating an] the measure- 
ment of zenith or nadir distances, is due to Professor 
Bohnenberger of Tubingen. 




Nl 



APPENDIX II. 



The Mural Circle. 



269 



1. — By means of the transit and a time-keeper, distances are meas- 
ured on the equinoctial in time ; and by an easy reduction this time is 
converted into arc. The object of the Mural Circle is to measure dis- 
tances on the meridian. 

This instrument consists of a metallic circle A A, varying in diameter 
from four to eight feet, strongly framed together or cast in one entire 
piece, and a telescope, of considerable optical power, having a focal length 
about equal to the diameter of the circle. The circle is firmly attached 




to the larger end of a hollow conical-shaped axis at right angles to its 
plane, which axis is mounted on Ys, placed in an opening through a 
heavy wall, whose front face is in the plane of the meridian. The gradu- 
ation is usually, though not always, upon the outer rim, and the readings 
are made by a pointer and six or more reading microscopes F, mounted 
upon the face of the wall, at equal distances from each other, around the 
circle. The telescope is mounted upon the front face of the circle, so as 



270 SPHERICAL ASTRONOMY. 

to move paiallel to the plane of the latter by means of a second axis, 
which turns freely and concentrically within that of the circle. The axis 
of the telescope is also conical, and is kept in place and proper contact 
with that of the circle, by means of a strong nut, which receives a screw 
cut upon its smaller end, the head of the nut bringing up against the end 
of the circle's axis. By turning this screw in the direction of its thread, 
the two axes are brought as closely in contact as may be found desirable. 

Permanently connected with each end of the telescope is a clamping 
arrangement, for the purpose of seizing the rim of the circle, and when 
these are in bearing, the telescope can only move with the circle, and 
when loose, it may move independently, thus affording the means of meas 
uring the same angular distance on different parts of the circle. 

Five clamping and tangent screw arrangements are permanently at- 
tached to the face of the wall, for the purpose of restricting the motion 
of the circle to the minute adjustments necessary to complete the contact 
of the objects observed with the reticle of the telescope, and to secure the 
instrument till the readings are made and recorded. They are made thus 
numerous, that one may always be at hand, in the various positions of the 
observer about the circle ; one of them is shown at E. 

The proportions of the whole instrument are so adjusted as to throw 
its centre of gravity on the axis just behind the circle, and between it 
and the wall, where the axis is received by a stirrup with friction -rollers 
C C, the stirrup being connected by rods D D with levers and counter- 
poising weights, which take the bearing from the Ys. 

The front Y, or that nearest the circle, is movable in azimuth about a 
vertical pintle, and that at the smaller end admits of both a vertical and 
horizontal motion, by means of two sets of antagonist screws. 

The tube of the telescope is perforated on the side opposite that of the 
axis, to admit the light from a lamp at a short distance in front of the 
circle ; this light is received upon a perforated reflector within, after the 
manner of the transit, and thrown to the eye to illuminate the field of 
view in nocturnal observations. The intensity of the illumination is reg- 
ulated by square perforations in two sliding plates, placed over the aper- 
ture in the tube, and so connected with rack and pinion work as to move 
in opposite directions, on turning a large milled-headed screw near the 
eye-glass ; one of the diagonals of each square being placed in the direc- 
tion of the motion of the plates, the figure of the opening will be un- 
changed, while its size may be varied at pleasure. 

At P and P are two small tubes, permanently fixed to that of the 
telescope, and at right angles to its length. They are cut away on one 



APPENDIX II. 271 

side at the middle, and each is closed at one end by a small disk of 
mother-of-pearl, movable about an axis perpendicular to its plane, and 
concentric with the tube. Between the disk and middle of the tube is a 
convex lens, which admits of a motion in the direction of the tube, and 
by which an image of a small eccentric perforation in the disk is formed 
about the middle of the cut, and of course on one side of the axis. A 
motion of the pearl causes this image to describe the circumference of a 
circle, of which the centre is on the axis of the tube. In the opposite 
end of the tube is a small microscope to view this image. The image is 
technically called the ghost, being a visible but unsubstantial representa- 
tion of the perforation. 

A small metallic style projects from the face of the wall at S, from the 
end of which may be suspended a plumb-line of fine silver wire, with its 
bob immersed in a vessel of water or other liquid at the bottom of the 
wall. The style is so arranged by an adjusting screw as to bring the 
plumb-line to intersect, the axes of the small tubes in the cuts, or to throw 
it clear of the instrument, at pleasure. 

In the tail end of the telescope, and at the solar focus of the object- 
glass, is a reticle, of which the axial wires are parallel to the axis of the 
circle. An additional wire is driven by a micrometer screw in the direc- 
tion, perpendicular to the axial w 7 ires, while it is also kept constantly par- 
allel to them. 

The telescope has a colli mating eye-piece, which is used for the same 
purpose and in the same manner as in the transit. 

*n Adjustments. 

2. — The adjustments are, first, to make the line of collimation per- 
pendicular to the axis, and, second, to make the axis perpendicular to the 
meridian. The plane of the circle and tube of the telescope are placed 
at right angles to the axis by the manufacturer ; the face of the wall is 
built as nearly in the meridian as possible by the aid of meridian marks ; 
and the Ys are so placed as to bring the axis, when mounted, nearly per- 
pendicular to the face, so that the adjustments are approximately made 
when the instrument is put up. To complete them, begin with 

3. — The line of collimation. — Turn the circle till the telescope is 
vertical, suspend the plumb-line and bring it by its adjusting screw to co- 
inci le with the upper ghost as seen through the microscope : examine 
the position of the lower ghost ; if it be not on the line, turn the pearl 
about its axis till it is : clear the line from the instrument, and invert the 
telescope by revolving the circle through 180°; bring the line tc the 



272 SPHERICAL ASTRONOMY. 

upper ghost as before, and again examine the lower ghost ; if it be on 
the line, the axis of the circle is horizontal, but if not, bring it to the 
line, one-half b) T the vertical adjusting screws of the circle's axis and half 
by a revolution of the pearl. When by repeating this process onr-e or 
twice the axis is made horizontal, put on the collimating eye-piece, and 
directing the telescope to the trough of mercury at the foot of the pier, 
and immediately below, move the diaphragm of the cross wires till the 
wire, which is perpendicular to the axis, coincides with its image — the line 
of collimation will be in a vertical plane, and of course perpendicular to 
the axis, which is horizontal. 

Should the telescope have no collimating eye-piece, recourse may be had 
to the vertical collimator, which is to be used exactly as in the transit. 

Since reflection takes place in a plane normal to the reflecting surface, 
the axis may be made horizontal by observing the same star directly, 
and by reflection from the free surface of mercury. If the time of the 
star's appearing on the line of collimation in both views be the same, the 
two positions of the line of collimation will lie in the same vertical plane, 
and being equally inclined to the horizon, the axis with which they make 
a constant angle must be horizontal. 

4. — Axis perpendicular to the meridian. — This adjustment may be 
made by the method pointed out for the same adjustment in the transit ; 
and when not perfected, the amount of error may be found by the process 
explained for that instrument. 

Polar and horizontal points. — On the circumference of the 
circle is a scale of equal parts, each part having an angular value of five 
minutes. Every twelfth division is numbered, the numbers varying from 
1 to 360° inclusive ; these indicate the degrees of the scale; and to facil- 
itate the reading, the intermediate divisions are also numbered, but in 
smaller characters. 

If the reading be known when the line of collimation is either hori- 
zontal or directed to the pole of the heavens, and the reading be taken 
when directed upon the centre of any body as it passes the meridian, the 
difference of the readings will in the first case be the observed meridian 
altitude of the body, and in the second its observed polar distance. 

5. — The horizontal point. — This is found by means of the collima- 
ting eye-piece, or vertical collimator, by the process indicated at page 268, 
or as follows, viz. : having carefully ascertained the value of a revolution 
of the micrometer in the eye-piece of the telescope, and the reading of its 
divided head when the movable wire is coincident with that parallel to 

he axis, set the telescope nearly in the position at which a star would 



APPENDIX II. 973 

appear by reflection on the stationary wire ; clamp the circle and record 
the reading of the index and microscopes ; when the star is at a conve- 
nient distance from the meridian wire, bisect it by the movable wire with- 
out moving the circle, and note the time accurately. Unclamp the circle, 
and bring the star by direct view accurately on the stationary wire, by 
turning the whole circle about its axis ; again note the time, and record 
the reading by the index and microscopes. Denote by R the first read- 
ing, by D the second, and by m the angular value of the distance between 
the fixed and movable wire, as indicated by the micrometer ; then, if the 
star had been observed accurately on the meridian, would the reading of 
the horizontal point be 

R ± m + D 



since the star must appear as far below the horizon by reflection as it 
actually is above it. But as the star cannot be taken at the same instant 
in both positions of the instrument, the readings R and D, taken as above 
indicated, must be reduced to what they would have been if taken on the 
meridian. 

6. — This correction will now be explained. 
Let S' SS" be the small diurnal circle Fig. 27. 

of the star ; P M S' an arc of the meridian ; 
S the position of the star when observed 
on the intersection of the axial and one of 
the side normal wires ; MS C the arc of 
a great circle, of which the axial wire is a 
portion. The point M will be that to 
which the line of collimation is actually 
directed, and ,5" is that in which the star 
will reach the meridian; the arc M S r is, 
therefore, the reduction to the meridian. 

Make P = MP S = hour angle of star ; 

d = P S — polar distance of star ; 

y = P M = polar distance of line of collimation ; 

x = M S f = reduction to meridian. 

Then in the triangle MP S, right-angled at M } 

sin y cos d 




cos P = tan y . cot d = 



and subtracting this from 



cos y * sin d T 



1=1, 

18 



274 SPHERICAL ASTRONOMY. 

we have, after reducing, and replacing 1 — cos P by 2 sin 2 \P, 

. „ _ _ sin (d—y) 

2 sin s AP= -— r~. 

cos y . sin a 

The observation being made very near the meridian, P and d—y will b« 
very small, and hence 

2 sin 2 \P = 2. (I? . sin l") 2 = \P 2 . sin 2 1"; 
sin {d—y) = sin x = x . sin 1" ; 
sin d . cos y = -J sin 2 67, very nearly. 

which in the above equation give, after reduction, 

x — isin 2 c?.P 2 . sin 1", 

in which P is expressed in seconds of arc. To express it in time, make 
P=15 P„ and we shall finally have 

225 . , „, . „„ 

x = . sin 2 d . P, 2 . sin 1", 

4 

P t denoting the number of seconds of time in the hour angle of the star. 

If, now, the numbers on the circle be supposed to increase in the direc- 
tion from the pole to the zenith, and the observed reading be denoted by 
P, then, since the line of collimation is nearer the pole than tht, place of 
culmination of the star, will the true reading be 

225 
P — x = P sin 2d. P, 2 sin 1" . . . . (21) 

4 v ' 

for all stars whose declinations are of the same name as the latitude of 
the place, and above the pole, and 

225 

P + x = B-\ sin 2 d . P, 2 sin 1" (22) 

4 

for all stars below the pole, or whose declinations are not of the same 
name as the latitude. 

7. — The interval P is obtained from the indications of a time- 
keeper. This usually runs too fast or too slow. To get the true from the 
indicated interval, suppose the time-keeper to gain or lose a seconds du- 
ring one revolution of the earth upon its axis. Itenote by A the number 
of sidereal seconds in the time of this revolution, and by t the true interval 
sought; then will 



APPENDIX II. 


A ±a 


:A 


: P : *, 


A 


P = 


1 


A±a' 


, a 

l± A 



275 



t = — -f-.P = .P; 



in which P is the indicated interval. 

Developing the coefficient of P, and limiting the series to the first 

power of — , because « is usually a small number of seconds, we have 

or replacing A by its value 86.400, 

t = (1 q= . 000012 a) P = a . P, 
in which 

a = 1 q= .000012 a. 

Substituting aP for P[ in equations (21) and (22), and making 

i = a? = 1 q- . 000023 «, 

there will result for the true reading 

°25 
P =f= ^ — . i . P\ sin 2 rf sin 1" .... . (23) 

8. — Denote by 2? and R the readings of the circle by the direct 
and reflected views ; by x and x the corresponding reductions to the me- 
ridian ; by m the small difference observed between the angle of incidence 
and reflection, and by H the reading of the horizonta point ; then will 



and 



D — a;±w+ R + x' D-\-R,m x—x' 
I£ = — • = dc , 

2 2 2 2' 

^• = ^±^d=^- — J.sin 1". sin 2 c* (P 3 ~P' 2 ) . (24) 



9, — Value in arc^ of units on the screw-head connected with the 
movable wire. — Run the movable wire to one edge of the field of view, 
say the upper, and bring it by a motion of the circle upon some well- 
defined and distant object; read the circle and micrometer; run the wire 
to the opposite or lower edge of the field, and by a motion of the circle 
bring the wire to same object again ; read the circle and micrometer as 
before, and divide the difference of the circle readings, reduced to seconds, 
by the difference of the micrometer readings, expressed in units of the 
screw-head ; the quotient will be the value sought. Or, 

Invert the telescope over a basin of mercury, by moving the circle, and 



276 



SPHERICAL ASTRONOMY 



Fig. 28. 



bring the image of the movable wire, supposed at one edge of the field, 
to coincide with the wire itself ; read the circle and micrometer : move 
the wire to the opposite edge, and turn the circle till the wire and its 
image again coincide, and read as before; divide the difference of the 
circle readings, reduced to seconds, by the difference of the micrometer 
readings expressed in units of its screw-head ; the quotient will be the 
value sought. 

Altitude and Azimuth Instrument. 
1. — This instrument, as its name indicates, is employed in the 
measurement of vertical and horizontal angles. It has two graduated 
circles and a telescope. The planes of the circles are at right angles to 
each other ; one called the azimuth circle, being connected with a tripod, 
by which it is levelled and kept in a horizontal position ; while the other, 
called the altitude circle, is mounted upon a horizontal axis, with which 
the telescope is also united, after the manner of the transit. 

To the centre of the tripod 
A A is fixed a vertical axis, of 
a length equal to about the 
radius of the circle ; it is con- 
cealed from view by an exterior 
cone B. On the lower part of 
the axis, and in close contact 
with the tripod, is centred the 
azimuth circle C, which admits 
of a horizontal circular motion 
of about three degrees, for the 
purpose of bringing its zero ex- 
actly in the meridian; this is 
effected by a slow moving- 
screw, the milled head of which 
is shown at D, This motion 
should, however, be omitted in 
instruments destined for exact 
work, as the bringing the zero 
into the meridian is not requi- 
site, either in astronomy or sur- 
veying : it is, in fact, purchasing 
a convenience too deaily, by 
introducing a source of error 




APPENDIX II. 277 

not always trivial. Above the azimuth circle, and concentric with it, is 
placed a strong circular plate F, which carries the whole of the upper 
works, and also a pointer, to show the degree and nearest five minutes to 
be read off on the azimuth circle ; the remaining minutes and seconds 
being obtained by means of the two reading microscopes F. This plate, 
by means of the cone B, rests on the axis, and moves concentrically with 
it. The conical pillars H support the horizontal or transit axis /, which, 
being longer than the distance between the centres of the pillars, the pro- 
jecting pieces c, fixed to their top, carry out the Ys a, to the proper dis- 
tance, for the reception of the pivots of the axis ; the Ys are capable of 
being raised or lowered in their sockets by means of the milled-headed 
screws b, for a purpose hereafter to be explained. The axis, with its load, 
is prevented from pressing too heavily on its bearings, by two friction- 
rollers, on which it rests ; one of these rollers is shown at e. A spiral 
spring, fixed in the body of each pillar, presses the rollers upward, with a 
force nearly a counterpoise to the superincumbent weight ; the rollers on 
receiving the axis yield to the pressure, and allow the pivots to find their 
proper bearings in the Ys, relieving them, however, from a great portion 
of the weight. 

The telescope K is connected with the horizontal axis, as before re- 
marked, in a manner similar to that of the transit instrument. Upon the 
axis, as a centre, and in contact with the telescope on either side, is fixed 
the double circle J. The circles are united by small brass bars ; by this cir- 
cle the vertical angles are measured, and the graduations are cut on a narrow 
ring of silver, inlaid on one of the sides, which is usually termed the face of 
the instrument : a distinction essential in making observations. The clamp 
for fixing, and the tangent-screw for giving a slow motion to the vertical 
circle, are placed beneath it, between the pillars H, and attached to 
them, as shown at L. A similar contrivance for the azimuth circle 
is represented at M. The reading microscopes for the vertical circle 
are supported by two arms bent upward near their extremities, and 
attached to one of the pillars. The projecting arms are shown at N 
and the microscopes above at 0, the latter admitting of a slight motion 
by means of antagonistic adjusting screws independently of the sup- 
porting arms. 

A reticle consisting of five equidistant axial and as many equidistant 
normal wires, is in the principal focus of the object-glass. The illumiin- 
tion of the wires at night is by a lamp, supported near the top of one or 
the pillars at d, opposite the end of one of the pivots of the axis, which, 
being perforated, admits the light to the centre of the telescope tube, 



278 SPHERICAL ASTRONOMY. 

where, falling on a diagonal reflector, it is reflected to the eye, and ilia 
mines the field of view. 

The vertical circle is usually divided into four quadrants, each num- 
bered 1°, 2°, 3°, &c, up to 90°, and following one another in the same 
order of succession ; consequently, in one position of the instrument alti- 
tudes are read off, and with the face of the instrument reversed, zenith 
distances ; and an observation is not to be considered complete till the 
object has been observed in both positions. The sum of the two readings 
will always be 90°, if there be no error in the adjustments, in the circle 
itself, or in the observations. 

It is necessary that the microscopes and the centre of the circle 
should occupy the line of its horizontal diameter ; to effect Avhich, an up- 
and-down motion, by means of the screws 5, is given to the Ys. A 
spirit-level P is suspended from the arms which carry the microscopes : 
this shows when the vertical axis is set perpendicular to the horizon. A 
scale, usually showing seconds, is placed along the glass tube of the level, 
which exhibits the amount, if any, of the inclination of the vertical axis. 
This should be noticed repeatedly whilst making a series of observations, 
to ascertain if any change has taken place in the position of the instru- 
ment after its adjustments have been completed. One of the points of 
suspension of the level is movable, up or down, by means of the screw f, 
for the purpose of adjusting the bubble. A striding-level, similar to the 
one employed for the transit instrument, and used for a like purpose, resfr- 
upon the pivots of the axis. It must be carefully passed between the 
radial bars of the vertical circle to set it up in its place, and must be re- 
moved as soon as the operation of levelling the horizontal axis is per- 
formed. The whole instrument stands upon three foot-screws, placed at 
the extremities of the three branches which form the tripod, and brass 
cups are placed under the spherical ends of the foot-screws. A stone 
pedestal, set perfectly steady, is the best support for this as well as the 
portable transit instrument.- 

Adjustments. 

2. — These have for their object to make, 1st, the azimuthal axis per- 
pendicular to the horizon ; 2d, to make the axis of the vertical circle 
horizontal ; 3d, to place the vertical circle at such a height that its mi- 
croscopes shall point to the opposite extremities of a horizontal diameter ; 
4th, to make the line of collimation perpendicular to the axis of the alti- 
tude circle, and horizontal when the reading of the vertical circle is zero. 

3. — The vertical axis. — Turn the instrument about, until the spin. 1 - 



APPENDIX II. 279 

level P is lengthwise in the direction of two of the foot-screws, when by 
their motion the spirit-bubble must be brought to occupy the middle of 
the glass tube, which will be shown by the divisions on the scale attached 
to the level. Having done this, turn the instrument half round in azi- 
muth, and if the axis is truly vertical, the bubble will again settle in the 
middle of the tube; but if not, the amount of deviation will show double 
the quantity by which the axis deviates from the vertical in the direction 
of the level ; this error must be corrected, one-half by means of the two 
foot-screws, and the other half by raising or lowering the spirit-level itself, 
which is done by the screw represented at/. The above process of rever- 
sion and levelling should be repeated, to ascertain if the adjustment has 
been correctly performed. 

Next turn the instrument round in azimuth a quarter of a circle, so that 
the level P shall be at right angles to its former position ; it will then be 
over the third foot-screw, which may be turned until the air-bubble is 
again central, if not already so, and this adjustment will be completed ; if 
delicately performed, the air-bubble will steadily remain in the middle of 
the level during an entire revolution of the instrument in azimuth. These 
adjustments should be first performed approximately, for if the third foot- 
screw is much out of the level, it will be impossible to get the other two 
right. The vertical axis is now adjusted. 

4. — The axis of the vertical circle. — This adjustment is performed 
exactly as in the transit, by means of the striding-level. 

5 .— Height of the vertical circle.— -The last adjustment being made, 
bring the microscopes to their zeros, and turn the vertical circle slightly, 
the striding-level being still mounted, till some one of its divisions be 
brought to the cross wires of one of the microscopes. Examine the other 
microscope, and if its cross be not on or near the division of the circle, 
180° distant from the first, depress or elevate the circle by the milled 
screws b till it is, keeping the axis horizontal by means of the level ; this 
will givt a sufficient approximation to bring the error of adjustment within 
the range of the adjusting screws which move the microscopes indepen- 
dently of their supporting arm. Recourse must now be had to these 
screws, by turning which in the direction indicated by the relative posi- 
tion of the circle division in question and the cross wires, the adjustment is 
perfected. 

0. — The line of colli mation. 

As the vertical circle is not, like the mural, generally used as a differen- 
tial instrument, but in the measurement of absolute altitudes or zenith 
distances, it is not only necessary that the line of collimation shall be per* 



280 SPHERICAL ASTRONOMY. 

pendieular to the transit axis, but also that it shall he parallel to the 
radius of the graduated circle drawn to the zero of its scale. 

Let x denote the angle made by the line of collimation with the plane 
normal to the transit axis, which angle is usually very small, and a the 
reading of the azimuth circle, when the telescope is pointed to some well- 
defined object in or near the horizon. If the line of collimation lie on the 
side of the normal plane, towards the zero of the circle, the true reading 
will be sensibly equal to 

a — x, 

if there be no other error of adjustment. 

Now revolve the instrument in azimuth 180°, bring the telescope again 
on the object, and denote by a' the new reading ; the true reading now 
will be 

a'-\- x; 

the difference of these true readings is obviously a semi-circumference, 

whence 

a — a f — 2x = 180°; 
and 

a— a' — 180° 



2 
and the true reading in the second position becomes 

a — a'— 180° 



«'+ 



Again, denote by y the small angle which the line of collimation makes 
with the plane passing through the axis of the vertical circle and that zero 
of this latter circle nearest the line of collimation, and suppose the line of 
collimation to lie above this plane when the telescope is directed to the 
same object, as before. Let b denote the apparent altitude, supposing 
the circle in the position to mark altitudes ; the true altitude is sensibly 
equal to 

6 + y.; 

turn the instrument in azimuth 180°, and bring the telescope again on the 
object ; the line of collimation will now be below the plane of the axis 
and zero, but the circle now indicates a zenith distance &', whence the 
trae zenith distance is 

adding these measures together, w r e have 



APPENDIX II 2SX 

b+ b'+ 2y= 90° 
90°_(6 + 6') 

y= 5 , 

and the true zenith distance becomes 

9Q°-(6 +S') 

Whence to adjust the line of collimation we have this rule, viz. : 

Direct the telescope to some well-defined and distant object, not far from 
the horizon, and bring its image to the intersection of the middle wires ; 
record the reading of the azimuth and vertical circles ; turn the instru- 
ment in azimuth 180°, bring the line of collimation again on the object, 
and record the new readings of the circles ; subtract from the difference 
of the azimuthal readings 180°, divide by 2, and add (algebraically) the 
quotient to the last azimuthal leading for a new reading in azimuth. Add 
the two readings of the vertical circle together, subtract the sum from 90°, 
and add half the difference to the last reading for a new reading on the 
vertical circle. Set the circles to these new readings, clamp, and by the 
adjusting screws of the reticle bring the line of collimation to the object, 
and the adjustment is made ; it should be verified, however, by repetition. 

7. — To make the normal wires perpendicular to the transit axie, 
proceed as in the case of the transit instrument, viz. : Move the diaphragm 
about in its own plane, till the image of some object appeai-s to run accu 
rately along some one of the wires, say the middle one, while the tele- 
scope is turned about its axis 

8. — The altitude and azimuth instrument is regarded by many as 
the most universally useful of all astronomical instruments. It is portable 
and accurate. When used in the meridian, it may perform the work of 
the transit and mural circle, though with somewhat diminished accuracy. 
But its principal merit consists in the ease with which it may be moved in 
azimuth without impairing its measurement of altitudes and zenith dis- 
tances. 

9. — The instrumental bearing of an object is the angle indicated 
by the reading of the azimuth circle when the centre of the object is ap- 
parently on the line of collimation. From the instrumental, the true 
bearing, or true meridian, is found by a process to be explained hereafter. 

10. — To find the altitude and instrumental bearing of an object at 
any instant, it is only necessary to make the object pass the line of colli- 
mation by turning both tangent-screws as it moves through the field of 
view, and to note the time of passage, ana wad the circles. 



2a2 SPHERICAL ASTRONOMY. 

11. — The altitude and time, or the instrumental bearing and time, 
are the elements more commonly observed in the case of celestial objects. 

12. — To obtain the altitude and time. — With the circles undamped, 
direct the telescope, which it will be remembered inverts, so as to bring 
the image of the object in the lower or upper part of the field of view, as 
the body may be rising or setting ; clamp the circles, and by the tangent 
screw of the azimuth motion, bring the image to the middle normal wire, 
and keep it there till it passes all the axial wires, carefully noting the 
time of its passing each, and also noting the indications of the level be- 
fore it passes the first and after it passes the last one. Now read the 
vertical limb, unclamp, and, by an aziinuthal motion, reverse the face of 
the vertical circle without unnecessary loss of time, and go through the 
same operation as before. Reduce the vertical readings to the same de- 
nomination of altitude or zenith distance, correct them by applying the 
level readings, and take half the sum for the altitude or zenith distance, 
as the case may be. Add the times together, and divide the sum by the 
number of recorded times for the corresponding time. 

13. — To find the instrumental bearing and time, direct the telescope 
as before, and clamp ; with the tangent-screw of the vertical motion, bring 
the image of the object to the middle axial wire, and keep it there till it 
passes all the normal wires, on each of which record the time. The read- 
ing of the azimuth circle will give the instrumental beaiing, and a mean 
of all the times will give the corresponding time. 

All of this supposes that the object's change in altitude and azimuth is 
uniform ; and although this is not strictly true, it is nevertheless so nearly 
so for the short time its image is in the field of view, that the error will 
be inappreciable during the interval required for a single set of ob- 
servations. 

The Equatorial. 

1, — The object of the equatorial or parallactic, as it is frequently 
called, is to support a telescope, generally of great size and optical power, 
in su ;h manner as to give to the observer the means of directing it with 
ease to any part of the heavens, and to measure at once the apparent hour 
angle and polar distance of a heavenly body. In the principles of its con- 
struction, it is like the altitude and azimuth instrument, but differs from it 
m the position of its axes, which, instead of being vertical and horizontal, 
are, when in position, respectively perpendicular, and parallel to the plane 
of the equinoctial. The first is called the polar, the second the declina- 
tion axis. It has two graduated circles, one securely attached to each 



APPENDIX II. 283 

axis ; the plane of one, viz., that attached to the polar axis, is parallel, and 
the other perpendicular to the equinoctial. The first is called the hour, 
and the second the declination circle. By a motion of the polar axis, to 
which the supports of the declination axis are attached, the declination 
circle may be made parallel to any assumed declination circle of the celes- 
tial sphere. The polar axis, always much loaded, is, in low latitudes, con- 
siderably inclined to the horizon, and the practical difficulty of supporting 
it has given rise to a variety in the form of the instrument. That repre- 
sented in the figure is the one now most generally used, and it is intro- 
duced here on that account. The principle is the same in all. 

The supporting-stand is shown at H, H, If. It is made either of a 
strong; frame of wood-work, or is cut from a solid block of stone. B is a 
plate of metal, fhmly secured to the stand, the surface of contact being 
parallel to the axis of the heavens. Upon this plate the instrument is 
mounted. The polar axis is seen at I. It is of steel, and revolves in two 
cylindrical collars near the extremities, and the lower end, being rounded 
off and highly polished, rests upon a steel plate attached to a bearing- 
piece K. 

To the lower end of this axis is attached the hour-circle is?, which is 
either graduated into hours, minutes, and seconds, or into degrees and the 
usual subdivisions, at the option of the person ordering the instrument 
The verniers, or reading microscopes, and tangent-screw arrangement, are 
supported by pieces connected with the plate B. The declination axis 
revolves in a metallic tube M, which forms a part of the frame-work se- 
cured to the top end of the polar axis. To one end of the declination axis 
is attached the declination circle P, which is graduated so as to read polar 
distances or declinations — suppose the former, its micrometers and tangent- 
screw being mounted upon pieces projecting from the extremity of the 
tube M, and to the other end, which projects slightly beyond the frame- 
work, is attached the telescope at a point nearer the eye-end than the 
middle. The excess of weight towards the object-end is, in the mounting 
by Mr. Henry Fitz, of New York, compensated by a counterpoise cylin- 
drical lever within the tube of the telescope, and so arranged in bearing 
as to counteract all tendency in the tube to bend. Attached to the end 
of the declination axis, is a counterpoise weight 0, the office of which is to 
throw the centre of gravity of the entire movable part of the instrument 
in the polar axis near its upper end, where it is received by a pair of fric- 
tion-rollers. 

At C is a box containing a system of wheel -work, so connected with 
the polar axis as, by the lid of weights and a centrifugal governor, to give 



284 



SPHERICAL ASTRONOMY 
Fig:. 29. 




it a uniform motion of rotation. The velocity of rotation is regulated by 
a vertical motion of the axis of the governor, whose balls in their retro- 
cession and increasing velocity, force a pair of rubbing surfaces against the 
interior of an inverted conical box : the moment of the friction thence 
arising equilibrates that of a descending weight, and the motion becomes 



APPENDIX II. 285 

uniform. By elevating the axis of the governor, the motion is ac<4lcr- 
ated ; by depressing the axis, it is retarded, and thus the velocity of rota- 
tion may be made equal to that of the earth about its axis, in which case 
a star in the field of view will be kept there ly the instrument itself, the 
effect being the same, abating refraction, as though the earth were at 
rest. 

2. — With a divided object-glass for the telescope, to be explained 
presently, or with the position micrometer, the equatorial is mostly used 
as a differential instrument, and particularly when the observer is pro- 
vided with a very full and accurate catalogue and map of the stars, which 
serve as points of reference. Whenever it is possible to bring a known 
object into the field of view with one that is not known, the place of the 
latter is found by measuring its bearing and distance from the known 
object. 

3. — To measure directly the hour angle and polar distance of an 
object with the equatorial, requires the parts of the instrument to be in 
perfect adjustment among one another, and its polar axis to be parallel 
to the axis of the earth. For these adjustments and a full analysis of the 
equatorial. 

Analysis of the Equatorial. 

The true instrumental position of an object is that indicated by an in- 
strument in perfect adjustment within itself. The apparent instrumental 
position is that actually indicated by an instrument whether in adjustment 
or not. When the several parts of an instrument are adjusted with respect 
to each other, these two positions are the same. 

The instrumental hour angle of an object, is its angular distance from a 
vertical plane passing through the polar axis, estimated upon the hour 
circle. 

Its instrumental declination is its angular distance from a plane perpen- 
dicular to the polar axis, estimated upon the declination circle ; and its in- 
strumental polar distance, its angular distance from the polar axis. 

The line of collimation should be perpendicular to the declination axis, 
and the latter perpendicular to the polar axis. The index of the hour cir- 
cle should stand at the zero of the scale when the line of collimation is 
parallel to the vertical plane of the polar axis, and, supposing the instru- 
ment to read polar distances, the index of the declination circle should be 
at the zero of its scale, when the line of collimation is parallel to the polar 
iixi«. 



es6 



SPHERICAL ASTEOXOMT. 



Supposing none of the conditions to be fulfilled, the apparent instru- 
mental position of an object will differ from the true, and the first thing 
to be done is to find the latter from the former, when the error in each 
of the above particulars is known. To do this, we will premise that the 
equatorial may be regarded as an universal transit instrument, whose 
horizon is the equinoctial, and zenith the pole. The formulae of reduc- 
tion applicable to the transit will apply at once to the equatorial by 
making therein the symbol for the latitude 90° ; in which case we shall 
have for the difference between the true and apparent instrumental hour 
angle in arc, the sum of the last three terms of Eq. (lo), viz., 



cos COS cos 



sin (X - S) 



which reduces, by making \ = 90°, and replacing 8 by 90° -- if, to 

c . cosec «r -f- I . cot if + z ; 

in which c is the error in the line of collimation, I that of the declination 
axis, and z a constant correction to be applied to every reading of the hour 
circle arising from the improper position of its index, and therefore the in- 
dex error of the hour circle, and <x the instrumental polar distance of an 
object whose image is on the line of collimation. 

Denoting by <r' the true, and by <f the apparent instrumental hour angle, 
and writing d <r for z, we have 

<r' = <r + ^c-f-Z. cot * -\- c . cosec if .... (a) 

Denoting by <ic' the true instrumental polar distance, and by d <k the in- 
dex error, then will 

it' == # + Jif (6) 

Let us next find from the hour angle and polar distance as given by the 
instrument, whose parts are in perfect adjustment among themselves, the 



true hour angle and polar distance, re- 
ferred to the true meridian and pole of n 
the celestial sphere. It is obvious that 
the instrumental and true co-ordinates 
would not differ, if the polar axis of the 
instrument were parallel to that of the 
heavens. Suppose this latter condition 
not fulfilled, and that the inclination of 
these axes is very small, as it always is, 
after putting the instrument in position 



Fig.l 




APPENDIX II, 287 

after the manner to be explained presently. Let P M be the arc of the 
meridian ; P, the true pole of the heavens ; P r , the point in which the 
polar axis of the instrument produced pierces the celestial sphere ; and S, 
the position of a star. Make 

p =z P S = the true polar distance ; 
it' = P' S = the instrumental polar distance ; 
s — MP S = the true hour angle of the star. 

The difference between the true and instrumental hour angles will be 
sensibly equal to the difference of the angles w T hich the true and instru- 
mental declination circles of the object make with the plane of the celestial 
and instrumental axes, or SPR — S P' R ; PR being P' P produced. 
Make 

X = PP'; 

<p == MP P\ positive when this angle is to the west of the meridian 

c = SPR] 

d = PSP'. 
Then in the triangle S P P' ', we have, Napier 's Analogies, 

cos h (p — if') 



tan £ (P + P f ) =*cot i d . 



cos £ (p ■+- if')' 



But P = 180° — c, and replacing cot \ d by its value in terms of sin and 
cos, we have 

L 2 v ,A 2 v ; sin i d cos £ (p + *') ' 

taking the reciprocal, and observing that the angles J (c — P'), \ d, and 
J (p — or') are very small, and that p = or', we have very nearly 

c — P' — s — a' = d . cos or' (c) 

But from the same triangle we have 

• j ,7 -v sinP 
sin or 

and this in Eq. (c), observing that the angle P = s — cp, gives 

s — d' == X . sin (s — <p) . cot or' ; 

transposing and replacing d f by its value in Eq. (a), we have, since s and 
tf only differ by a small quantity, 

s = tf + ^c + X sin (tf — 9) cot nt + c . cosec * + / cot or . . (c?) 

With £ as a pole, and radius S P\ describe the arc P' T y then will 



288 SPHERICAL ASTRONOMY. 

PS=p = «' + PT. 
But within the limits supposed 

P T = X . cos (s — <p) = X . cos (tf — (p) ; 
whence, replacing if' by its value in Eq. (6), we have 

p = it + 4 * + X . cos (rf — 9) (e) 

Adjustments. 

The adjustments of the equatorial are of two classes, viz. : those which 
relate to the parts among one another ; and those which determine the 
position of the instrument in relation to the celestial sphere. 

The rules for the first are suggested by equations (a) and (6), and are 
as follows : 

Index Error of the Declination Circle. — Direct the line of collimation 
to any well-defined object in any part of the horizon ; in reversed positions 
of the declination circle, the readings of this circle in Eq. (b) give 

v r = it + Jit; 
<k' = * t — ^ <x. 

Taking the second from the first, 

j <g — 1 



Apply this with its proper sign to the last reading <g f , and the telescope 
still being upon the object, move the verniers or microscopes till they indi- 
cate this corrected reading. 

Line of Collimation. — The preceding correction being applied, move 
the telescope till the declination circle marks a polar distance equal to 
90° ; then by a motion of the polar axis, bring the line of collimation upon 
some object directly in the instrumental east or west ; read the hour cir- 
cle ; reverse the declination circle ; bring the telescope upon the same 
object, and read again ; and these readings, in Eq. (a), will give, since 
*r = 90°, 

c / = a + A a + c, 

<t'=(t t — 12 h + ^tf — c; 

whence, subtracting the second from the first, 

tf— (f,+ 12 h 
c=- . 

Apply this to the last reading tf„ and move the instrument about its polar 



APPENDIX II. 2S9 

axis till the vernier indicates this reading ; then by a motion of the adjust- 
ing screws which act upon the telescope, bring the line of collimation tc 
bear upon the object. 

Declination Axis. — Turn the line of collimation to an object directly in 
the instrumental north or south, to get the greatest declination. This wijl 
give to I its greatest effect. Read the hour circle as before in the direct 
and reversed position of the declination circle. Then, since by the last 
adjustment c = 0, we have 

<f' = <f + A <f + I . cot «r, 

tf' = <S 4 — 12 h -f- A a — I cot * ; 

whence, by subtraction and reduction, 

<f-(f + 12 h " 
I = -- . tan if ; 

or, if the telescope be set to a polar distance equal to 45°, 

Set the hour circle to the last reading tf,, corrected by the above value 
of 7, and bring the line of collimation back to the object by the adjusting 
screws, which act upon the declination axis. 

The Polar Axis parallel to the Axis of the Heavens. — About the time 
that some circumpolar star, the nearer the pole the better, comes to the 
meridian — say its upper passage — turn the declination circle till it reads 
the star's polar distance, increased by the refraction due to its altitude, and 
clamp the declination circle ; then by a motion of the entire instrument in 
right ascension, and the screws which act upon the polar axis in the me- 
ridian, bring the star to the cross wires, and keep it there till the instant, 
as indicated by a time-piece, of its crossing the meridian. This will be 
sufficient for the first approximation. 

Then observe some well-known star in quick succession very near the 
meridian, reversing the declination circle. The reading of the declination 
circle, corrected for refraction, will give, in Eq. (e), since tf = T 

p = if -f- A if -\- X . cos 9, 
p — yf< — A if -f- X . cos <p ; 
whence 

if -f tf 4 
X , cos 9 = p — -. 

The first member being the projection of the arc X on the meridian, is the 

19 



290 SPHERICAL ASTRONOMY. 

arc by which the pole is too high or too low. The axis being moved 
through this distance by estimation, direct the telescope to the polar dis- 
tance, corrected for refraction, of a second star soon to come to the me- 
ridian ; when the star is in the field put the clock movement in motion, 
and as the star culminates, as indicated by a time-piece, bring the axial 
wire to the star by the adjusting screws of the polar axis which are in the 
meridian. 

The polar distance of another star when six hours from the meridian 
being observed in quick succession, in the direct and reversed position of 
the declination circle, Eq. (e) gives, since in this case <S = 90°, 

p = <g + 4 it + X . sin 9, 
p = it t — d v + X sin <p ; 

whence 

. . if + if, 
X . sm 9 = p -. 

The first member is the projection of the arc X, on the declination circle at 
right angles to the meridian, and is, therefore, the deviation of the pole of 
the instrument from this latter plane. This error being treated in a man- 
ner similar to the preceding, by means of adjusting screws which act at 
right angles to the meridian, the polar axis is brought to this latter plane, 
and the instrument will be so nearly in ' adjustment as to bring the errors 
within the limitations required to render equations (d) and (e) exact. 

The approximation may be continued, if desirable, or the value of each 
error found by recourse to celestial objects properly selected, and these 
errors employed as elements of reduction. 

To find c. — Observe an equatorial star about the time of its meridian 
passage, and again after reversing the declination circle ; the readings of 
the hour circle in Eq (d) give 

s = <J-\-4<f-\-c cosec if, 

s f = a' — 12 h + A a — c cosec * ; 

whence 

c = ^ J - i '- sin if. 

2 

Denote by a the right ascension of the star, and by t and f the sidereal 
times of observation, we have 

s=t— -a; s' = t' — a; 

these in the above equation give 



APPENDIX II. 291 

(t-t r ) - (rf- <r') - 12* 



sin* . . . (/) 



To find I. — Observe some star, near the pole, in quick succession revers- 
ing the declination circle ; the readings of the hour circle in Eq. (d) give 

* = <f -f- J a -f- X sin (tf — 9) cot if + c . cosec if + I cot if, 

s' = <r'— 12 fa + ^f tf + X sin (tf' I2 h — 9) cot cr — c . cosec t — I cot * ; 

whence, since the third terms of the second members do not differ sensibly, 

(s - s') - (<f — <f') - I2 h 

I . cot if -f c cosec if = *- ; 

eliminating s and s' by their values I — a, and <' — a, and reducing, 



r* — t' — tf — a' — 12 h ] . sin it — 2 c , v 

2 . cos AT v/ 

To find 9 cmrf X. — Observe any well-known star, and again after revers- 
ing the declination circle. The readings of the circles in Equations (d) 
and (e) give 

8 — <f + A C + X . cot if . sin (tf — 9) + », 
/ = 0" — 12 h -f A a + X . cot if sin {i' — 12 h — 9) + n\ 
p =if -f- A if -f X . cos {d — 9), 
y = flr, + Aflr+X.cos (<j"— 12 h — 9) ; 
in which 

n = c . cosec if + I . cot if, 
n r =c . cosec tf, ■+• I . cot #,, 

Subtracting the first from the second, the third from the fourth, transptn 
sing, and making, after eliminating s' and s by their equals t' — a, t — a, 

2 = (t r - t) - (tf' - tf - 12 h ) - (»' - »), 

n-=(p , -^ J p) .-(*,-*), 

we obtain 

X , cot if [sin (0" — 12 h — 9) — sin (tf — 9)] = 2 > 
X.[cos(fl" — 12 h - 9) - cos(j'~9)] = n S ' ' W 

but 

«in( ff '- 12 h -^)- sin((r-.^) = 2siti|(ff'- <r-12 h ).cos ( "„"" " 12 - <p), 

j 1 1 I oh v 

cos(cr'- 12 h - 0) - cos (<r -0) = 2 sin £(*'-*- 12'') < lin ( L!!1 ^ - <P) I 



292 SPHERICAL ASTRONOMI. 

substituting these above, and dividing the second equation by the first, we 
have, using p for *r, 

C + tf - 12 h 



tan 



\ n 

~ V = ¥ cot ? • • • • (*) 

whence 

<r'-f(r-i2 h , n 

9 = tan" 1 . — cot p . . . . (k) 

and from equations (A) we have 

in is 



. , , N . /V+ff— 12* \ us /(r '_l_ ff _12h \ 

smi(ff — ff— 12h). siul <p) siru(V— (r— 12 h ). cosl $y 



(I) 



To find 4 ft. — Observe a star before its culmination in the hour angle 
360° — tf, and at an interval after its culmination in the hour angle <f\ 
such, that 360° — ft and tf / shall be equal, or very nearly so, without re- 
versing the declination circle ; Eq. (d) will then give 



24 h — s = 24 h — ft + A ft + X cot * sin (360° — tf + 9) — », 
s' = o" -f- A tf + X cot # . sin (o" — 9) + n. 

Adding and reducing, 

s' — s = a" — ft + 2 A ft + X cot * . [sin (<j" — 9) — sin (tf + 9) ] ; 

writing sin (tf — 9) for sin (ft' — - 9), to which it is sensibly equal, we have, 
after developing the last term, reducing, and replacing s and 8 r by their 
equals t — a and V — a, 

A # — \ 1 1 L _}_ x . cot it . (sin 9 . cos ft) . . (m) 

z 

For a star in or near the equator, we may take cot ie = ; or for a 

star whose hour angle is 90°, in which case cos ft = 0, the above value for 

index error becomes 

If - t) - (a' -ft) 
A<t — ± ' ? '- (m') 

2 V / 

To find Air. — Observe the same star twice in quick succession, and in 
reversed positions of the declination circle ; the readings of the declination 
circle, in Eq. (e), give 

p = K + A if + X cos (rf — 9), 

p = <g' — A if + X cos (<t — 9) ; 

whence by subtraction, 

*' — ir 
A* = — ........ (n) 



APPENDIX II. 



Heliometer. 



1. — The image formed by a lens of a point on he surface of an ob- 
ject, is on a line drawn through the optical centre of the lens and the 
point. If the point be stationary and the lens in motion, along a line 
perpendicular to this line, the image will also be in motion, and in the 
same direction. 

Every fragment cut from a lens by a section parallel to its axis, forms 
an image just as large and as perfect as does the entire lens, the only 
difference being in the intensity of its illumination, which will be less in 
proportion as the surface of the fragment is less than that of the en- 
tire lens. 

Ifj then, the lens be divided by a plane through its optical axis, and the 
two halves moved in opposite directions, and perpendicular to this axis, 
an image of an object formed by the entire lens will be duplicated, and 
the individuals of the pair will be equally bright. Two half lenses, so 
mounted as to be moved parallel to the dividing plane, called the plane 
of section, and at light angles to the optical axis, by means of micrometer 
screws, constitute the Heliometer. Such an arrangement forms the object- 
glass of the telescope at Z, in Fig. 29. The screws are furnished with 
large circular heads, which are carefully graduated after the manner ol 
those of the position micrometer, and are turned by the aid of a rod, 
reaching to the eye-end of the telescope. The entire frame-work, which 
supports the slides of the semi-lenses, admits of a rotary motion about 
the axis of the telescope's tube, and is put in motion by a second rod, 
also passing to the eye-end. By this last arrangement the plane of sec- 
tion may be made to pass through any two objects, whose images are 
simultaneously in the field of view. 

2. — The value in arc of the linear distance through which the 
images of the same object are made to separate, by turning the microm- 
eter screw-head through each unit of its scale, is found by a process in all 
respects similar to that explained in Appendix No. I., for the position 
micrometer. 

3. — Directing the telescope to the sun, duplicating its image, and 
turning the micrometer screws till the images are tangent, the leading 
multiplied by the angular value of the head unit will gi e the apparent 
diameter of the sun. Hence the name of the instrument. 



294: 



SPHERICAL ASTRONOMY 



4. — But it is obvious that the apparent dimensions of any othei 
body may be measured in the same way. Also the angular distance sub- 
tended by the line, joining two objects, whose images may be brought 
into the field of view together. For this purpose, turn the whole field lens 
till the plane of section pass through the objects, duplicate the image of 
both, and turn the micrometer screws till one of the images of the one 
be brought to coincide with an image of the other ; the reading, treated 
as before, will give the angular distance sought. 

The Sextant. 



1. — This is employed in the measurement of the angular distance 
between two objects. It is one of the most generally useful instruments 
that has yet been devised, furnishing, as it does, data for the solution of a 
variety of astronomical problems of the greatest practical utility both on 
tand and at sea. It is especially useful at sea, where the unstable position 
of the mariner excludes the use of almost all other instruments. 

It depends upon this catoptrical principle, viz. : When a ray of light 
is reflected by two plane reflectors in a plane normal to both, the ray 
is devi.ited or bent from its original direction through an angle equal to 
twice the angle made by the reflectors. 

Let A C and C B represent the section of two plane reflectors perpendic- 
ular to their line of intersection C. RM, MN, and JV the course of a 
ray reflected first at the point M, and next at the point iV; then will the 
angle R h' -2ACB. For, draw the 
normals MD, M' D' to the reflector A C, 
and DD' to the reflector C B, and denote 
by 9 and 9" the angles of incidence on the 
reflector AC: by a' that on the reflector 
C B, and by i the inclination of the reflec- 
tors. Then, since by the principle of optics 
the angle of incidence is equal to that of re- 
flection, we have from the triangle M D N, 



Fig. so. 



9 = *; 




and from the triangle M' D' i\ r , 



adding these, we have 



9 — 9 '= ^; 



9 — 9"= 2i 



and because MD and M' D' are parallel, the first member is the inclina- 
tion of the first incident to the second reflected ray. 



ATPESTDXS'II. • 395 

If then the reflector CB were transparent at the point N, the waves of 
ight from an object at R', would be transmitted through it and coincide in 
direction with those from R reflected at M and iV; and to an eye situated 
at 0, the objects R and R' would apparently coincide. Two reflectors so 
mounted as to give the means of reading their inclination to each other, 
when this coincidence takes place, would give the angular distance R R r 
of the objects by simple inspection ; and, with appliances to facilitate the 
operations of the observer, constitute a reflecting instrument, which, ac- 
cording as its arc of measurement is extended to an entire circumference or 
limited to an arc of 90°, 60°, or 45°, is called a reflecting circle, quadrant, 
sextant, or octant. The sextant is the more common of the instruments 
with limited arcs now in use. 

2. — The annexed figure represents a sextant. It consists of the two 
plane-glass reflectors C and B seen edgewise; a graduated arc A A, of 
which the plane is perpendicular to those of the reflectors ; an index-arm 
F, vernier V, clamp and tangent screw ; a telescope ED, of which the 
line of collimation is parallel to the plane of the arc of measurement; col- 
ored glasses L and iif to qualify the light received into the telescope, and 
a triangular system of frame-work uniting strength with lightness, to sup- 
port all the parts and render them available. The handle of the instru- 
ment is represented at H. 

The arc of measurement is divided into half-degree spaces, which are 
numbered, as whole degrees, and these divisions are subdivided to any de- 
Fig. 81. 




296 • SPHERICAL ASTRONOMY. 

sirable extent consistent with facility of reading. The reflector B, called 
the index-glass, is covered with an amalgam of tin on the face towards the 
eye- end of the telescope, and turns with the index-arm about an axis in its 
own plane, and through the centre of the arc < £ measurement, being per- 
pendicular to the plane of the latter. The reflector C, called the horizon- 
glass, is, abating the limited range of the adjusting screws, securely fixed 
with its plane also at right angles to that of the arc of measurement. Only 
one-half of this glass is covered, and that half lies nearest the frame of the 
instrument, the covered face being turned from the telescope. The line 
separating the covered from the uncovered part of this glass is parallel to 
the plane of the graduated arc, and at a distance therefrom about equal to 
that of the line of collimation, being sometimes a little greater and some- 
times a little less in consequence of a change in the position of the tele- 
scope, to make the supply of light it receives through the uncovered, equal 
to that which enters it after reflection from the coated part of the horizon- 
glass. The position of the telescope is altered by means of a screw and 
milled nut connected with its supporting ring U. By turning the nut the 
telescope is thrust from or drawn towards the face of the sextant. This 
device is called the up-and-down piece. There are usually six or seven 
colored glasses of different shades, which are so mounted that they can be 
turned about an axis c or b parallel to the face of the sextant, and be inter- 
posed or not at pleasure. 

To facilitate the reading, a small microscope G is attached to a swing 
movable about an axis a, connected with the index-arm. Two telescopes 
and a plane tube, all adapted to the ring U, are packed with the sextant 
One of these telescopes has a greater magnifying power than the other, and 
inverts the visible images of objects. The telescopes are provided with 
colored glasses, which are so mounted as to be easily attached to the eye- 
end to qualify the light of the sun when that body is observed. 

Adjustments. 

3. — The sextant requires three adjustments, viz. : 1st. To make the 
index and horizon glasses perpendicular to the plane of the arc of measure- 
ment. 2d. These glasses parallel to each other when the index is at the 
zero of the scale. 3d. The optical axis of the telescope parallel to the 
plane of the arc of measurement. 

4. — To accomplish the first, move the index to the middle of the arc, 
then holding the instrument horizontally with the index-glass towards the 
eye, look obliquely down this glass so as to see the circular arc by direct 
view and by reflection at tho same time. If the arc appear broken, the 



APPENDIX II. 207 

position of the glass must be altered till it appear continuous, by means of 
small screws that attach the frame of the glass to the instrument. 

The horizon-glass is known to be perpendicular to the plane of the in- 
strument when, by a sweep of the index, the reflected image of an object 
and the image seen directly, pass accurately over each other ; and any er- 
ror is rectified by means of an adjusting screw, provided for the purpose, at 
the lower part of the frame of the glass. 

5. — The second adjustment is effected by placing the index or zero 
point of the vernier to the zero of the limb ; then directing the instrument 
to some distant object (the smaller the better), if it appear double, the ho- 
rizon-glass must, after easing the screws that attach it to the instrument, if 
there be no adjusting screw for the purpose, be turned around a line in its 
own plane and perpendicular to that of the instrument, till the object ap- 
pear single ; the screws being tightened, the perpendicular position of the 
glass must again be examined. The adjustment may, however, be rendered 
unnecessary by correcting an observation by the index error. The effect 
of this error on an angle measured by the instrument is exactly equal to 
the error itself: therefore, in modern instruments, there are seldom any 
means applied for its correction, it being considered preferable to determine 
its amount previous to observing, or immediately after, and apply it with 
its proper sign to each observation. The amount of the index error may 
be found in the following manner : clamp the index at about 30 minutes 
to the left of zero, and looking towards the sun, the two images will ap- 
pear either nearly in contact or overlapping each other ; then perfect the 
contact, by moving the tangent-screw, and call the minutes and seconds 
denoted by the vernier, the reading on the arc. Next place the index 
about the same quantity to the right of zero, or on the arc of excess, and 
make the contact of the two images perfect as before, and call the minutes 
and seconds on the arc of excess the reading off the arc; half the differ- 
ence of these numbers is the index error; additive when the reading on the 
arc of excess is greater than that on the limb, and sublractive when tb« 
contrary is the case. 

Example. 



Reading on the arc 
" off the arc 


. . 31 56 

, . . 31 22 


Difference . . . 


34 


Index error . . . 


= — 17 



298 SPHERICAL ASTKONOJkT. 

In this case the reading on the arc being greater than that on the arc 
of excess, the index error, = 17 seconds, must be subtracted from all ob- 
servations taken with the instrument, until it be found, by a similar pro- 
cess, that the index error has altered. One observation on each side of 
zero is seldom considered enough to give the index error with sufficient 
exactness for particular purposes : it is usual to take several measures each 
way ; " and half the difference of their means will give a result inore to be 
depended on than one deduced from a single observation only on each 
side of zero." A proof of the correctness of observations for index error is 
obtained by adding the above numbers together, and taking one-fourth of 
their sum, which should be equal to the sun's semidiameter, as given in 
the Nautical Almanac. When the sun's altitude is low, not exceeding 20° 
or 30°, his horizontal instead of his perpendicular diameter should be 
measured (if the observer intends to compare with the Nautical Almanac, 
otherwise there is no necessity) ; because the refraction at such an altitude 
affects the lower border (or limb) more than the upper, so as to make his 
perpendicular diameter appear less than his horizontal one, which is that 
given in the Nautical Almanac : in this case the sextant must be held 
horizontally. 

6. — The third adjustment is made by the aid of two parallel wires 
placed in the common focus of the telescope for the purpose of directing 
the observer to the centre of the lield of view, in which an observation 
should always be made; these wires are parallel to the plane of the instru- 
ment, and divide the field of view into three nearly equal parts. The sun 
and moon are made tangent to each other, when their angular distance is 
90° or more, at one of the wires ; the position of the sextant is then altered 
so as to bring these bodies to the second wire; if the contact continue, the 
line of collimation is parallel to the plane of the instrument ; if not, the 
position of the telescope must be altered by means of two adjusting screws 
connected with the up-and-down piece. 

Artificial Horizon. 

1. — To measure directly the altitude of any celestial object with the 
sextant, it would be necessary that the object and horizon should be dis- 
tinctly visible ; but this is not always the case in consequence of the irreg- 
ularity of the ground which conceals the hcrizon from view. The observer 



APPENDIX II. 
Fig. 82. 




"XS' 



is therefore obliged to have recourse to an artificial horizon, which consists 
usually of the reflecting surface of some liquid, as mercury contained in a 
small vessel A, which will arrange its upper surface parallel to the natural 
horizon DAC. A ray of light S A, from a star at S, being incident 
on the mercury at A, will he reflected in the direction AE, making the 
angle S A C= C A S' (AS' being E A produced), and the star will ap 
pear to an eye at E as far below the horizon as it actually is above it 
Now with a sextant whose index and horizon glasses are represented at 1 
ind R, the angle S E S f may be measured ; but SES r =SAS' — ASE, 
and because A E is exceedingly small as compared with A S, the angle 
AS E may be neglected, and S E S' will equal SA S', or double the alti- 
tude of the object : hence one-half the reading of the instrument will give 
the apparent altitude. At sea, the observer has the natural or sea horizon 
as a point of departure, and the altitude may be measured directly. 

8. — Having now gone through the principle and construction of the 
sextant, it remains to give some instructions as to the manner of using it. 

It is evident that the plane of the instrument must 
be held in the plane of the two objects, the angular 
distance of which is required. The sextant must be 
held in the right hand, and as loosely as is consistent 
with its safety, for in grasping it too firmly the hand 
is apt to be rendered unsteady. 

When the altitude of an object, the sun for instance, 
is to be observed, the observer, having the sea-horizon 
before him, must turn down one or more of the dark 
glasses uy shades, according to the brilliancy of the object; and directing 
the telescope to that part of the horizon immediately beneath the sun, and 



Fig. 83. 




3U0 SPHERICAL ASTRONOMY 

holding the instrument vertically, he must with the left hand slide the 
index forward, until the image of the sun, reflected from the index-glass, 
appears in contact with the horizon, seen through the unsilvered part of 
the horizon-glass. Then clamp, and gently turn the tangent-screw, to 
make the contact of the upper or lower limb of the sun and the horizon 
perfect, when it will appear a tangent to his circular disk. When an arti- 
ficial horizon is employed, the two images of the sun must be brought into 
contact with each other. To the angle read off apply the index error, and 
then add or subtract the sun's semidiameter, as given in the Nautical Al- 
manac, according as the lower or upper limb is observed, to obtain the ap- 
parent altitude of the sun's centre. 

The Principle of Repetition. 

1. — By this principle, the invention of Borda, the error of graduation 
in any instrument may be diminished, and, 
practically speaking, annihilated. Let P Q 
be two objects which we may suppose 
fixed, for purposes of meie explanation, 
and let L be a telescope movable on 0, 
the common axis of two circles, AML 
and a be, of which the former AM L is 
fixed in the plane of the objects, and car- 
ries the graduations, and the latter is free- 
ly movable on the axis. The telescope is 
attached permanently to the latter circle, 
and moves with it. An arm a A carries 
the index or vernier, which reads off the 
graduated limb of the fixed circle. This arm is provided with two clamps, 
by which it can be temporarily connected with either circle, and detached 
at pleasure. Suppose, now, the telescope directed to P. Clamp the index- 
arm A to the inner circle, and unclamp it from the outer, and read ofl 
Then carry the telescope round to the other object Q. In so doing, tho 
inner circle, and the index-arm which is clamped to it, will also be carried 
round, over an arc AB, on the graduated limb of the outer, equal to th< 
angle P Q. Now clamp the index to the outer circle, and unclamp tru: 
inner, and read off: the difference of readings will of course measure the 
angle P Q', but the result will be liable to two sources of error — that of 
graduation and that of observation, both of which it is our object to get 
rid of. To this end transfer the telescope back to P, without unolamping 
the arm from the outer circle ; then, having made the bisection of P, 







\_ 



~^) 



APPENDIX II. 



fry i c 



101 



clamp the arm to b, and unclamp it from B, and again transfer the tele- 
scope to Q, by which the arm will now be carried with it to C, over a 
second arc B (7, equal to the angle P Q. Now again read off ; then 
will the difference between this reading and the original one measure twice 
the angle P Q, affected with both errors of observation, but only with the 
same error of graduation as before. Let this process be repeated as often as 
we please (suppose ten times) ; then will the final arc AB M read off on the 
circle be ten times the required angle, affected by the joint errors of all the 
ten observations, but only by the same constant error of graduation, which 
depends on the initial and final readings off alone. 

The Reflecting Circle. 

1. — The use of this instrument is, in general, the same as that of 
the sextant ; but when it unites, as it often does, to the catoptrical prin- 
ciple of this latter instrument, the principle of repetition, it becomes, in 
the hands of a skilful observer, one of the most refined and elegant of 
the portable implements in the service of astronomy. 

This form of the instrument is represented in the annexed figure. 

The arc of measurement, which is extended to the entire circum- 
ference, is divided into 720 equal parts, and, for the reason explained 
in the account of the sextant, these parts are numbered as whole de- 
grees, the subdivisions being continued to any desirable degree of mi- 
nuteness. 

The circle is mounted upon two concentric axes, which may move in- 
dependently of each other, and also of the circle. Upon one end of the 




302 SPHERICAL ASTRONOMY. 

central axis is mounted a reflector E, similar to the index-glass of the, 
sextant, and upon the other an arm A C, in the position of a diameter of 
the circle. Upon the corresponding ends of the other axis are mounted 
a system of frame-work and a second arm BD. This frame-work sup- 
ports a second reflector F, similar to the horizon-glass of the sextant, a 
telescope H, colored glasses L and L\ and the handles 7", J, K for hold- 
ing in different positions. The reflectors are perpendicular to the plane 
of the circle. Each of the arms A C and B D has a vernier at both 
ends, and at one end a vernier, clamp, and tangent-screw, so that the re- 
flectors may be clamped in any position consistent with their being per- 
pendicular to the plane of the circle, and for each position there will be 
two arc readings, differing by 180°. 

At G is seen the barrel for the up-and-down piece, of which the milled 
head is concealed beneath the end B of the arm BD. 

At M are seen the microscope and its reflector for reading, mounted 
upon a pin projecting from the vernier arm. 

The circle is usually accompanied by a stand, to which it may be at- 
tached, when great steadiness is required, by means of screw holes in the 
handles ; one of these holes is seen in the handle /. 

Adjustments, 

2. — The adjustments are the same as those of the sextant, and 
performed in the same manner, with the exception of the index error, 
which, in this instrument, is always eliminated by the manner of ob- 
serving. 

Mode of Observing. 

3. — First Method. — The instrument being in adjustment, clamp the 
index A, and record its reading, noting the degrees, minutes, and seconds 
on the vernier A, and the minutes and seconds on the vernier C. Un- 
clamp the index B, and directing the telescope to one of the two objects 
whose angular distance is to be measured, move the whole circle around 
till the two images of this object are brought nearly together ; clamp the 
index B, complete the contact or coincidence by the tangent-screw, and 
record the reading as before, noting the degrees, minutes, and seconds on 
the vernier B, and the minutes and seconds on the vernier D. The 
glasses are now parallel. Unclamp A, and, holding the circle in the 
plane of the objects, direct the telescope to the fainter of the two, and 
move the index A till the image of the second object is brought nearly 



APPENDIX II. 80/5 

in contact with that of the first ; clamp, and complete the contact by 
the tangent-screw : read the verniers A and C as before. The difference of 
the A readings will give the angle as measured by the sextant, and this 
angle should always be noted as a check. Next, unclamp B, and keeping 
the telescope upoia the same object, move the whole circle till the two 
images of this object are again nearly in contact ; clamp, and finish the 
contact by the tangent-screw. The glasses are again parallel, and the 
index B has passed over an arc equal to the angular distance of the two 
objects. Unclamp A, and move it in the same direction as before till 
the two objects again appear nearly in contact ; clamp, and complete the 
contact with the tangent-screw ; the index of A will thus have passed 
over an arc equal to twice the angular distance of the objects. Now un- 
clamp B, and turn the whole instrument as before till the two images of 
the same object again appear ; clamp, and complete the contact, and the 
index of B will also have passed over an arc equal to twice the angular 
distance of the objects. This process being repeated as often as may be 
deemed desirable, finally read the verniers as before. Take a mean of 
the minutes and seconds of the first reading of A and C, as also of B 
and D ; these with the degrees of A and B will give the true readings 
of the instrument at the beginning of the operation ; do the same for the 
last reading, or that, at the close of the repetitions. Take the difference 
between the last and first readings of the instrument for each set of ver- 
niers ; add these differences together, and divide the sum by the numbef 
of times that A and B have been moved after the first contact of the im 
ages of the same object : the quotient will be the angle sought. 

A comparison of this angle with that given by the difference of the 
second and first readings of A, will indicate the error, should one have 
been committed, either in the readings or in taking account of the num- 
ber of repetitions. 

Second Method. — Clamp A, and record the readings of A and as 
before ; unclamp B ; direct the telescope to the fainter of the two 
objects, and turn the circle till the second object appear nearly in contact 
with the first ; clamp B ; complete the contact by the tangent-screw, and 
record the reading of B and D. Now, invert the instrument by revolv- 
ing it through an angle of 180° about the line of collimation of the tele- 
scope ; unclamp A, and move this index till the objects again appear nearly 
in contact ; clamp, and complete the contact by the tangent-screw ; the 
difference of the second and first readings of A will be double the angu- 
lar distance of the objects, the half of which will be the check. Bring 
the instrument back to its former position by revolving it about the line 



30tt SPHERICAL ASTRONOMY. 

of collimation ; anc'iamp B, and turn the circle till the images again 
appear ; clamp, and complete the contact by the tangent-screw ; the arc 
passed over by B will also be double that of the objects. This process 
being repeated as often as the observer pleases, finally read the instrument 
on both sets of verniers ; take the first reading of A and C from the last ; 
do the same for B and D ; add these differences together, and divide the 
sum by twice the number of times that A and B have been moved since 
the first, contact. 

4. — The process of repeating is much facilitated by the following 
device. A brazen arc is attached to the frame-work of the instrument so 
as to be concentric with the arc of measurement, and just below it, and 
moves with the telescope and horizon-glass. It is out of view in the po- 
sition of the instrument represented in the figure. To this arc are fitted 
two small sliders, that adhere to it by friction, wherever placed. Firmly 
attached to the tangent-screw end of the arm A C are two small pieces of 
metal, called checks, lying in the direction of radii, and just long enough 
to cross the brazen arc, and to slide over its surface, after the manner that 
the index moves over the arc of measurement, so that if one of the sliders 
be interposed, the motion of the index will be arrested. 

5. — In the first method of observing, after the two images of the 
same object are made to coincide, place one of the sliders against the 
check on the side from which the index A must be moved to bring the 
other object in the field of view ; after the contact of the two objects is 
perfected, by moving the index A, place the other slider in contact with 
the other check on the opposite side. Now, the circle being in the plane 
of the objects, a little consideration will make it manifest, that to restore 
the contact of the images of the same, and afterwards of the two objects, 
it will only be necessary to bring the checks in contact with their respec- 
tive slides by alternately moving the circle and index A. The brazen arc 
is sometimes graduated and numbered in opposite directions, commencing 
from the positions of the checks, corresponding to the parallel position 
of the reflectors ; this furnishes an additional check upon the angle meas- 
ured, and facilitates the management of the sliders. The use of the sliders 
in the second method of observing is, from what has been said, too obvi- 
ous to need explanation. 

6. — This Appendix contains, it is believed, an account of all that 
is essential in the theory, construction, and use of the principal instru- 
ments employed in astronomical measurements. To describe ali that 
are in use, would expand the work to dimensions inconsistent with its 
object, viz. : to give to students in the threshold, as it were, of Astro- 



APPENDIX III. 



305 



nomy, a preparation for future progress in the subject. The German 
Meridian Circle combines the Mural and Transit, as does also the 
English Transit Circle. One of the most useful instruments to which 
the student can give his attention, is the Zenith Telescope, alluded to 
on page 199 of the text. 



APPENDIX III. 



ATMOSPHERIC REFRACTION. 

When light passes from one medium to another it is refracted according 
to the law expressed by the equation, Optics, § 15, 

sin z = m sin z' . (1) 

in which z is the angle which the normal to the inci- 
dent wave makes with the normal to the deviating 
surface, and z r the angle which the normal to the de- 
viated wave makes with the same. 

Denote the angle of deviation SAS r by r, then 
will 

z r = z — r; 

which substituted in Eq. (1), we have, after develop- 
ing, 

sin z = m (sin z cos r — cos z sin r) ; 

and because r is always small when m differs little from unity, which is 
the case in the passage of light through the different strata of the atmo- 
sphere, we may write 

cos r = 1, and sin r = r ; 

and dividing both members of the above equation by cos s, we have 

m — 1 







. tan z 



(2) 



and regarding z as constant, r will vary with m, and hence 



dm 
dr — — r tan z 

mr 



(3) 



Now, if we regard the atmosphere as composed of indefinitely thin and 
concentric strata of increasing density from the top to the bottom, dr will 
be the deviation of the ray in passing from one stratum to another whose 

20 



306 



SPHERICAL ASTRONOMY. 



indexes of refraction differ by dm. But in the same kind of medium, this 

difference is found by careful experiment to be directly proportional to the 

difference of densities ; hence 

dm , „ 

— = \dD, 

m* 

in which X is a constant and D the density of the atmosphere at the place 
of any one stratum whose index is m. Whence, Eq. (3), 



dr = \ . dD . tan z 

Let B be the arc of the earth's surface in 
a vertical plane through a heavenly body S ; 
0' N' and 0" N" , two concentric strata of 
atmosphere in the same plane; S 0" 0' 0, 
the curve which is normal to the front of a lu- 
minous wave coming from the body to the 
observer at 0. As an object is seen in the 
direction of the normal to the wave from it as 
the wave enters the eye, the body will appear 
to an observer at to be at S\ on the line 
tangent to the curve at ; and if SP be the 
prolongation of the straight portion of the ray 
before it enters the atmosphere, the angle 
S T S' will be the total deviation. This angle 
S T S' is called the refraction. Denote the 
radius of the earth C by unity ; the height 
of the stratum N' 0' above the surface by x\ 
the angle CO' by 6. Then taking 0' and 0" 
contiguous, we have 0' CO" = d&, MO" = 
dx\ and the angle of incidence HO" S, on 
the stratum of which 0" N" is the upper lim- 
it, being denoted by 0, we have M 0' = dx 
tan z t and 

dx tan z 



d& = 



1 +x 



(4) 



Fig. 8. 




(5) 



And denoting the apparent zenith distance Z' S' by Z, we have 

r=TPZ' -Z = 6+z~Z; 
and by differentiating 

dr = d& +dz\ 



APPENDIX III. 



or substituting the value of dQ, in Eq. (5), 
whence, Eq. (4), 



' dx . tan z _ 



or 



■> j r» * tfa; . tan s 

Xa Z> . tan z = f- as ; 

1 -\- x 

'■ =Xd£ 



by integration, 



tan z 1 -f- x ' 

log sin S == Xi) — log (1 -|- #) -|- (7; 

and making # = 0, in which case D = D J and z = Z, we have 

logsin£=Xi>, + C; 
and by subtraction, 

sin z ' _ . 



or 



whence 



But, Eq. (4), 



log 
log 



sin Z 


= - K W ~ 


^j- 


- lOg {L -f ^ 


<h 


sin z 


= log*-* 00* 


-D) 


- log (1 + 


x); 


sin Z 


sin Z : 


sin Z . e~~ A 

1 + Z 


(A- 


J>) 










dr- 


\dD tan z = 


X sin ^ . d D 






Vi 


— - sin 2 z ' 





(6) 



and substituting the value of sin z above, 

X.^nZ.e- x ^- D KdD 
dr = —— = = =- ■••(') 

V(i +x y- s in 2 Z.e- 2 ^ D '-^ 

If the law which connects the varying density D with the height x be 
given, one of these variables may be eliminated and the integration per- 
formed. But in a practical point of view this is not necessary ; for X is 
known to be a very small fraction, as is also the greatest value of x, the 
latter not exceeding 0.01931, being the height of the first stratum of air 
that has sensible action upon light, divided by the radius of the earth, or 
77 miles divided by 4000 miles. Developing the factors e~~ ■ '~ ' and 
e~ ^ ~~ ', neglecting the second and higher powers of X and .r, and 
also the term of which X sin 2 Z is a factor, which may be done without 
sensible error when Z does not exceed 80°, we find 



308 SPHERICAL ASTRONOMY. 

X. sin ZdD \smZ.dD 



dr = 



Vl + 2a; -sin 2 Z vW Z + 2x' 



whence 



X . tan Z . dD „ , . _ , 

dr = = X tan Z . (1 - x sec 2 Z) dD ; 

Vl + 2zsec 2 Z 



r = X tan zf[dD - sec 2 Zztf D) ; 
and performing the integration, that of the last term by parts, 
r = X tan Z \D - sec 2 Z (Dx —fl> -dx)\\ 

but if A denote the height of the mercurial column at any stratum of air 
above the observer, D n the density of the mercury, and g the force of 
gravity regarded as constant, then will 

g.fDdx = 9 D u h; 

and 

r = X tan Z [D - sec 2 Z (Dx - D u h) + G\\ 

and from the limit x = 0, where D = D' and h = h t , to the limit x = 

height of the entire atmosphere, where D = 0, r = 0, and h = 0, we find 

r - X tan Z . D' (l - A . ^f' sec' A 
Taking the density of Mercury as unity, we have the mean value of 
The mean value of h is found from the proportion, 

miles inches 

4000 : 29.6 : : 1 : h: 
■which will give for the coefficient of sec 2 Z, 

h>-jjr = 0.0012517. 

Also, if D i be the density of air when the thermometer is 50, and the ba- 
rometer 30 inches; and we take a = 0.00208, and j3 = 0.0001001, the 
coefficients of expansion for air and mercury respectively, then, Analytical 
Mechanics, § 245, 

A l + (50-Q ./3 
' " 30 ' 1 -f (i — 50) . a " 

in which t denotes the actual temperature of the air and mercury supposed 
the same, and h the height of the barometer. Hence 



APPENDIX III. 309 

r = \D . A . 1±$™JZAI . tan Z (1 - 0.0012517 sec 8 Z) . . (8) 

' 30 l+(«-50)a v / w 

Had the second power of # been retained in Eq. (7), then would 



A_ l+(50-«)* 

r AX/,. 



(2+sin* Z\ , , 
1—0.0012517 sec 2 Z+0.00000139 — ._ 1(8)' 
cos 4 Z / 

the last term of which, within the limits supposed, is insignificant. 
Make 

u = i ! 1 + ( 50 " g . tan Z . (1 - 0.0012517 sec 2 2) . . (9) 
30 1 + (^ — 50) a 

and we have 

r = \D,u . (10) 

Denote by z and z' the greatest and least observed zenith distances of a 
circumpolar star, r and r' the corresponding refractions, and c the zenith 
distance of the pole ; then will 

z -\- r -\- z' -\- r' 
C = 2 • 

In like manner, if z t and z/ be the greatest and least zenith distances of 
another circumpolar star, r, and r/ the corresponding refractions, 

„ _ z , + r , + z ! + T ! 

C ~ 2 ' 

Equating these values, replacing the refractions by the values given in Eq. 

(10), we find 

XD = Z < + S- - (g ~ *') . 
' w + «t' — (tt y — m/)' 

The indications of the barometer and thermometer being substituted in Eq. 
(9), give u, u', u 4 , and «/, and therefore the value of \D r Numerous 
and careful observations make \D t =. 5 7 ".82, which substituted in equa- 
tions (8) and (8)', give the refraction for every observed zenith distance, 
temperature of the air, and height of the barometer. 



__^ 



310 



SPHERICAL ASTRONOMY. 



APPENDIX IT 



SHAPE AND DIMENSIONS OF THE EARTH. 




'T 



Let A MP t A' represent a meridional 
section of the terrestrial ellipsoid, M the 
place of the spectator, B the north pole 
of the earth, C its centre, Z the zenith, 
HM H' a parallel to the rational horizon 
and tangent to the meridian section at M, 
A' A the intersection of the equator by 
the meridian plane. 

Make 

I = the angle MGA = PKH= latitude of M ; 
A = CA, the equatorial radius ; 
B = OB, the polar radius. 

Then, referring the curve to the centre and axis, its equation is 

A*f + B 2 x* = A*B* (a) 

the equation of the tangent line HE 7 , 

A*yy' + B 2 xx f = A 2 B* (b) 

and the equation of the normal at M, 

A*y' {x-x')-B*x'{y-y') = . . . , . (c) 

in which x' and y' are the co-ordinates of M. 

Denote the angle MTCbj T, then from Eq. (b) we have 



tan T 



B 2 x' 

A*y' 5 



but T=90° — I whence 



J3V tan l = A*y' 
Also, denoting the eccentricity by e, we have 



(<*) 



» 



Substituting x'y' for xy'm Eq. (a), combining the resulting equation with 
Eq. (d), and eliminating B by means of Eq. (e), we find 



APPENDIX IV 
A cos I 



VT 



sin 8 I 



A (1 — e 2 ) . sin Z 

y = — x ' 

Vl - e 2 sin 2 I 



Differentiating the first, regarding x and I as variable, we have 



311 



• (/) 



(1 - e 2 sin 2 Vj* 



(.?) 



but, designating by s the linear dimension of any portion of the arc of the 
curve, we have for the projection of the element ds on the axis of x, 

ds . cos T= ds . sin I; 

and since a? is a decreasing function of the latitude, 

— dx' = ds . sin I; 

which substituted in Eq. (g) gives 



(1 — e 2 sin 2 /)2 



For any other latitude I \ we have 



ds' = A . 



(1 -e 2 ) .dV 



(1 - e 2 sin 2 Z')2 
dividing the first by the second, making 

dl = dV = 1°, 
and solving with respect to e ? , aye find 

ds — ds f 



(A) 



(A)' 



From Eq. (h) we have 



3 ' ds sin 2 / -c/5' sin 2 V 
ds 



(1 - e 2 sin 2 If 



1 - e 2 

and from the well-known property of the ellipse, 
B = A VT^7 . 



.... (i) 
.... (/) 

. . . . (*) 

Making ds = c, ds' =c r , I = l mi V — l' m we have equations (10) and 
(11) of the text. 



312 SPHERICAL ASTRONOMY. 

Denoting by R the radius of curvature at any point of the meridian, we 
have 

dxd'y 

finding the values of d *, c?y, and d 2 y from Eqs. (/), and substituting 
above, there will result 

1-e* 



B = A 



(1 - e 2 sin 2 l)i 



Then 
whence 



2«R : 360° : : jS : 1°; 

(1 — e* sm J Z) 2 

in which {3 denotes the linear dimension of one degree of latitude. 

Denoting by p the radius of the earth in any latitude /, we obtain by 
squaring and adding Eqs. (/), 



p = 4. < / 1 __^____ r (m) 

Every section of the terrestrial spheroid through the centre is an ellipse of 
which the semi-transverse and semi-conjugate axes are respectively A and 
p, I being the latitude of the extremity of the conjugate axis. Denoting 
by e t the eccentricity of the elliptical section, we have 

2 _ ^ 2 - P 2 _ e 2 (1 -e 2 ) sin 2 ^ 
€ ' ~ A 2 ~ 1 - e 2 sin 2 I ' 

this value of e? substituted in Eq. (I) after making therein I = 90°, and 
denoting by $ t the length of a degree on the section perpendicular to the 
meridian in the latitude I, 



R — 2 * A A/ * — e * sin2 J , x 

P ' ~ 360 T 1 - e 2 (2 - e 2 ) sin 2 I ' ' ' ' W 

The value of the radius of the parallel of latitude is given by that of #', 

Eqs. (/) ; and denoting by a the linear length of a degree of longitude on 

this parallel, we have 

2tf.2rf jl cos Z ■ x 

a = . a = . A — .... (o) 

360 360 y/\ _ e 2 s i n 2 l 



APPENDIX V. 313 

Dividing both members of Eq. (a) by A 2 B 2 , making A = 1 and B =y, 
that equation becomes 

£ + *=! (P) 

7 

Differentiating, we find 

dx 1 y 
dy ~y 2 ' x' 

but the angle at M in the evanescent triangle mMhk equal to the angle 
at G — V in the triangle MGD\ and denoting in future the central lati- 
tude MCD by I, we have 

dx _. 

— = tan I . 

dy 



- = tan I ; 
x 



whence 



tan I = y 2 tan J' (g) 



Making ii = 1, B ==y, and eliminating e 2 from Eq. (m) by the relation 
£ = 1 — y 2 , we have 

P=-y===== • • • • W 

4/l+i-/-.sin 2 Z 
7 



APPENDIX V. 



EARTH'S ORBIT. 



The sun's attraction for the earth varies inversely as the square of the 
distance. The earth describes, therefore, an ellipse about the sun, having 
the latter body in one of its foci. 

By Eq. (266), Analytical Mechanics, we have 

da. 2c ,\ 

n = v w 

in which a denotes the angle which the radius vector of the earth makes 
with any assumed axis, r the radius vector, c the area described by the lat* 
ter in a unit of time, and t the time. 



314 SPHEEICAL ASTRONOMY. 

Also, Eq. (277), Analytical Mechanics, 

«d-0 = ^ s (b) 

in which a is the semi-transverse axis of the earth's orbit, e its eccentricity, 
and k the intensity of the sun's attraction on the unit of mass of the earth 
at the unit's distance. 

The polar equation of the ellipse is 

1 + e cos V ' ' - ' ' ' * • ' K c ) 

in which V is the true anomaly, estimated from the perihelion. 

Eliminating r and c from Eq. (a) by means of Eqs. (b) and (c), we have 

<JT. 3 

1^ ,dt = (l- e*) 2 . (1 + e cos V)- 2 .da ... (f) 
a* 

developing the factors of the second members by the binomial formula, 
and neglecting all the terms involving the powers of e higher than the sec- 
ond, we have 

—^ . dt = (1 — | e 2 ) (1 - 2e cos F+ 3c 2 cos 2 V— &c.) da 
a* 

and because 

cos 2 V= \ + \ cos 2 V, 

da = dV; 

and, § 201, Analytical Mechanics, 



Vh 2 - 



= jr = ™ W 



in which T is the periodic time, and m the mean daily motion of a point 
on the radius vector at the unit's distance from the sun ; whence we have 

mdt = da — 2e cos Vd V + | e 2 cos 2 Vd 2 V— &c. ; 

and by integration, 

m£+(7=a — 2esin F+f e 2 sin 2F- &c. 

Making V = 0, and estimating a from the line through the vernal equinox, 
we have 

mt p + C= a p ; 



APPENDIX V 315 

in which a p 's the longitude of the perihelion, and t p the time from peri- 
helion passage. Whence, by subtraction, 

m (t — t p ) = a — a p — 2 e sin V + f e 2 sin 2 V — &a , (e) 

but 

a-a p =F; 
whence 

m (t — t p ) = V-2esin F+ f e 2 sin 2 F— &c. . . (#) 

in which m (t — ^) is the mean anomaly, being the mean angular dis- 
tance from perihelion. 

Adding a p to each member of Eq. (e), making 

m(t- t p ) + a p = a m 

and writing a — a p for F, we find 

a m = a — 2 e sin (a — a p ) + J e 2 sin 2 (a — aj — &c. . . (A) 

m which a m is the mean, and a the true longitude. 

Denote by L the mean longitude at any given epoch, say the beginning 
of the year, and by t the interval of time since the epoch ; then will 

a m = L -f m t, 
and 

L + m t = a — 2 e sin (a — a p ) + | e 2 . sin 2 (a — a p ) — &c. . . (i) 

Again, assuming 

cos u — e 

cos V = (j) 

1 — e cos u x ' 

and substituting in Eq. (/), and replacing the first factor by its value in 
Eq. (d), we have 

mt -\- C ' = / du (1 — e cos u) = u — e sin u\ 

and making t = t p , in which case the body is : n perihelion, where V = 
and therefore u = 0, we have 

m* p + (7=0; 

and by subtraction, making £ — t p == t\ 

mt' z=u — e sin u (k) 

From Eq. (j) we have 

tan 2- = * / r3i- tan 2 ; 



316 SPHERICAL ASTRONOMY. 

from which we have 

K = tt + «smw + -.sin2w . . . . . (I) 

and from Eq. (&), 

u = m t' -f 2 e sin m t' + J e 2 sin 2 m 2' + &c. . . . (w) 

which substituted in Eq. (I) gives 

V=mt' + 2e sin mt' + %e 2 .sin 2mt f . . . . (n) 
whence 

7— wW'== 2e sin mi' 4- J e 2 . sin 2mt' . . . . (o) 

The first member, which is the difference between the true and mean 
anomalies, is called the equation of the centre. It is expressed in terms of 
the eccentricity and mean anomal}\ The auxiliary angle u is called th« 
eccentric anomaly. 



APPENDIX VI. 

PLANETS' ELEMENTS. 
Differentiating the equation 

e cos v = 1, 

r 

we have, after dividing by d t, 

dv L dr 

e sin v . — = — . -T- ; 
d t r dt 

but, Analytical Mechanics, § 192, 

dv 2c 
Tt = ~7' 

which substituted above gives, after making 

— - V 

dt~ n 

esmv = — . r r ; 
which is Eq. (100) of the text 



APPENDIX VII. 317 

APPENDIX VII. 

PLANETS' ELEMENTS. 



Prom Eq. (277), Analytical Mechanics, we have 
whence, making jx = Jc, 



4e 2 





2c 


— Vp 


. Ya (1 - 


-V); 


and this in the equatic 
Appendix VI., gives 


n 

dv 
dt 


dv 

Tt 


2c 






Va(l- 

r 2 


T 


and substituting the value of r 2 from the equation 






a 


(I-* 2 ) 




we find 

dt - 


3 


1 ■ 

(1 -c 2 


-f- € COS V 
3 

) 5 


dv 




"(1 + 


e cos ff) r 


To integrate this, assume 









cos u — e 
cos v = 



1 — c cos w ' 



from which find the value of dv; eliminate dv and cos v above, and we 
have, 

3 

dt = -— . (1 — e cos u) du ; 
V> 

and by integration 

3 

a? 
t+C = — — (u — e sin it). 



But, Analytical Mechanics, § 201, 



a* T 



in which T is the periodic time. 



318 SPHERICAL ASTRONOMY. 

Whence, making t = when u = 0, we have (7=0, and 

— . t = u — e sin u ; 

and denoting the mean motion by n, we lave 



2« 
n = 

and finally 



«= y; 



w t = u — e sin u ; 

which is Eq. (106) of the text. 

The quantity 2 is the time from perihelion, for by makiig u = 0, we 
have 

t = ; cos v = 1, or v = 0°. 



APPENDIX VIII. 

PLANETS' ELEMENTS. 

Differentiate the equation 

r 2 = # 2 + y 2 -J- g 2 , 
and divide by 2 re? £, we have 

dr x dx y dy z dz 
dt r ' d t r ' dt r'dt 



and making 



we have 



f*I - ir • dx -V • d JL-V > *±-V - 



*- = ■*+? 5*1^ 



which i&Eq. (112) of the text 



APPENDIX IX. 319 

APPENDIX IX. 

PLANETS ELEMENTS. 
Make 

or,, a 2 , a s , the observed right ascensions ; 
5„ (3 2 , [3 5 . the observed north polar distances ; 

*i» hi ^ tne niean times of observations reduced to any first meridian, say 
that of Greenwich ; 

and suppose the observed quantities corrected to the mean equinox and 
mean position of the equator at the beginning of the year. 

In the interval of time required for light to travel from a roaming body 
to the earth, the body describes some definite portion of its path, and at 
any given instant we see the place it left and not that which it actually 
occupies. We look, as it were, at luminous places on the orbit, but always 
behind the body's true place. The position which a body occupied at the 
instam the light started, and in which it is seen at a given time, is called 
its virtual place at that time ; and that which it actually occupies is called 
its true place. 

Conceive three sets of parallel rectangular co-ordinate axes, one set- 
through the place of observation, another through the centre of the earth, 
and the third through the centre of the sun. Take the planes xy parallel 
to the plane of the equinoctial, the axes of x parallel to the line of the 
equinoxes and positive towards the first point of Aries. 

Denote by p y the distance of the body's virtual place from the earth at 
the time t x , and by v the time required for light to travel over the mean 
radius of the earth's orbit, which we have taken as unity ; then will v p ; be 
the time required for light to travel over the distance p,. 

Denote by x, y, z the co-ordinates of the virtual, and %, y, z the co-ordi- 
nates of the true place of the body at the time t u referred to the centre of 
the earth ; then, regarding the motion of the body as uniform during the 
time v p„ will 

dx \ 



't'-Tt 



- dy 
y = y-v fl .- 

- dz 



(i) 



SPHERICAL ASTRONOMY. 



Denote the co-ordinates of the sun, cleared of aberration at the time t x , 
and referred to the same origin, by X„ Y t , Z t ; and the heliocentric co 
ordinates of the true place of the body at the same time by x { , y n z t ; then 
will 

x = X / + x„ 

y=r,+ y t , 
z = z , + z i ; 

which in Eqs. (1) give 



x = X t + x, 



V P, 



y = Y , + y, - v ?s 



*= Z i+ S i-V?i' 



dt 

djfi + y) 

dt 

d{Z t + z t ) 
dt 



in which 



P, = (*, + 2,) wc fr .... 
or, which may be preferable, if the body be near the equator, 
p / = (x t -f X) sec a, cosec /3, . . . 



(2) 



(3) 



w 



Denoting the co-ordinates of the virtual place of the body at the time t h 
referred to the place of observation, by x\ y\ z' ; ' and the co-ordinates 
at the same time of the place of observation, referred to the centre of the 
earth, by/„ g t , and A; then will 

y = y' + 9» 

z =z r + h; 
which substituted in Eqs. (2) give 

y' — x r tan pt, = ) , g . 

z' — x' tan d| — J ■•••••• w 



But 



in which 



APPENDIX IX. 

cotan 6 1 = cos a t . tan ft 



321 
■■'(*) 



Also, if I denote the geocentric colatitude of the place of observation, p the 
corresponding radius of the earth, and T t the sidereal time of observation 
reduced to degrees, then will 



nnd 



f t = p . sin I . cos T t 
g t = p . sin I . sin T t 

h = p . cos I . 



(8) 



sun's horizontal parallax at the place of observation 



number o. 



in an arc equal in length to radius 



Multiplying the first of Eqs. (5) by tan a, and subtracting the product 
from the second, then by tan & x and subtracting the product from the third, 
and reducing by the relations of Eqs. (6), we have 

y 1 - a5l tan« 1== (X 1 -/ 1 )tari ai -Z i + ^ + vp ] [§ + ^-tan ai (§ + -J i )] 

In like manner 

y -*a tan « 3 = (X 2 — / 2 ) tan a 2 — T 2 + g 2 + m[~^ + -jf — tan-a a (-^- + -jf)~] 



'(10) 



and 

gr s - - «a tan « 3 = (X 3 — / 3 ) tan a 3 — F 3 -f g z + ^[-J- 3 + — ^ — tan o 3 ^ -f -^)] 

^-^ tan , 3 = ( x,-/ 1 ) te n.:-^ + *+^[f+^- to n, 3 (f +§»)]; 

in which, as in equations (3) and (4), 

pg = (zg + Z 2 ) sec ft ) 
p 3 = (z$ + Z 3 ) sec ft f 

or, if the body be near the equator, 



(ii) 



Now make 



p 2 = (# 2 -f X 2 ) sec a 2 . cosec ft ) 
p 3 = (x 3 + X 3 ) sec a s , cosec ft ) 

*i = k — iy and * 3 = *, + r' ; 
21 



(12) 



SPHERICAL ASTRONOMY. 



then, because x x and x z are functions of t, which become x % when r and r' 
become zero, we have by Taylor's formula, 



dx 2 d 2 x<> t 2 

X 9 . T A . — 

2 dt + d* 2 2 



and the same for y t and ^ ; 



d 3 x 2 



d A 



d? ' 6 + dt* ' 24 &C * 



(13) 



dx 2 . d 2 x 2 <r" , d*x 2 <r' 3 , d*x 2 r' 4 , . 



and the same for y 3 and z 3 . 



The intervals <r and t' must be such as to make these expressions con- 
verge rapidly, and it will rarely if ever be necessary to retain the terms of 
the series involving powers of t and r' higher than the fourth. 

Denote by jia the acceleration due to the sun's attractive force at the 
mean distance of the earth, and by r 2 the distance of the body from the 

Mi 

sun at the time k, then will -r be the acceleration due to the sun's attrac- 



tion on the body, and we shall have 



d*x 2 

~d¥~~ 




x 2 


tix 2 * 

r 3 


dt" 






Mi 


d*z % 

~d¥ ~ 






\IZ 2 

,3 
r 2 J 



(14) 



Differentiating and dividing by d t, twice, we have 

<Px a ju, dx 2 3/x dr 2 

T? = " 7} ' It + T* ' It ' * 2 ' 



12 w, drf 



d i x r __p* 6fA dr a dx 2 Sfj, <Pr 2 jl^j* t*r, 



and the same for 



d*y 2 d A y 2 <Pz 2 d% 

dt*' dt> d? m* 



by simply writing y and z successively for x. 



APPENDIX IX. 



323 



These values being substituted in Eqs. (13), and the resulting values of 

*t, y»» z u ** v* and z s'i 

and also those of 

dx x dy x ds x dx % dy % dz t 

Tt' ~dt' It' ~dt' It' ~dt' 

obtained therefrom in Eqs. (10) — observing to limit the series to the secona 
order of differentials in the aberration terms, or those of which v is a factor, 
and to the fourth order in the others — we have, after making 

A t = (X x -f x ) tana, - F, + g x } 

B x = (X, -f x ) tan 4, - Z t + k 

A = (X 8 — f t ) tan a* — F 8 -f g 2 

B 2 = (X 2 -/ 2 ) tan d s - Z t + h 

A 3 = (X 3 — f 3 ) tan a s - F 3 + g 3 

B 3 =(X 3 -/ s ) tan 4* - Z 3 + h J 

, <*F, . rfJT, 

#, = — : tan a, — — - 

dt x dt 

_ <£ ^i . d X, 

c£F 2 dX, 

^ 2 = ___ toa2 _ 



^ 2 

n? 8 



rf Z* d X« 

— tan L —r- 

dt a dt 

dY 3 dX s 

__tana 3 — 



<£Z 3 rfX 3 



(15) 



(16) 



2r 1 ' t 2r 2 r dt + 2r/' rf^ 



fx t 4 c? 2 r 8 



^T' 



W = 



jxt 4 dr z 



JXT 3 



U' = 



w = 



ff.r' 



jut, r n d r, 



u.V 4 



GrJ 



2 r/ ^ "*" 2 r 2 s * c?** 8r/' rf^ 1 " 24r 4 « 
/at' 4 o?r 8 
477' "dt 



(17) 



324 



al= ( U+ ^f\ (y 2 -z 2 tan a x 



SPHLRICAL ASTRONOM 



.v p! )(g 2 -tan ai ^ + ^/ 



(is; 



a f =vp 1 ^ + -j F -tana 2 — J 
/_ dz 2 , d# 2 \ 

^ = (^-^)(y a - as fan^)+(W"+»p l )(^-tan« 3 ^+»pA 

6,= ( V- l ^fyz s -^n6 3 ) + (W' + ^p t -tan0 3 ^J +vp s / J 

and in which, by substituting the values of z, and S3, ar, and # 3 in equations 
(3), (4), (11), and (12), 

p 2 = (z 2 + Z 2 ) sec /3 a 

or 

pi = [z 2 (l - |~ s ) ~~~Jt ' r + Xl ~\ sec ai ' coseC ^ l 

p 2 = (# 2 + ^2) s ^c a 2 . cosec /3 2 

Ps = [^2(1 - Yp\ +-JI '*' + X s] sec a 3 . cosec /3, 



(19) 



(20) 



we obtain 



c?y 2 c?# s 

y 2 — # 2 tan a, — — — r -+- tan a L — r = ^ -+- a x 

1 — rr 2 tan 0, 7-? r + tan & x -^ r = B x -\-b x 

at at 



y t — x a tan a 2 
z» — x 2 tan 0, 



= 4> + a 2 
= -Bf + h 



dy« dx 2 . 

y 9 — x+ tan a 8 + -f- r' — tan a 3 -7- r = ^ 3 + a, 
ou at 



2, — s s tan d, f -j- <r' — tan & 3 -y- r = 



^ + 63 



(21) 



APPENDIX IX. 



325 



dx 2 dy« , dz 2 . ,;,. , 

Regarding # 2 , y iy z 2 , — , -f-, and — as unknown quantities, we ob- 



tain by elimination, 

in which, by making 
E 



* 2 = (P+P)-(Q + <1) 



(22) 



(<r'\ sin (a 2 — a,) cos a 3 *\ 
r/" sin (a t — a 3 ) cos a 2 

S = (i + -\ sin (*i - <M cos d 3 
\ r) '' sin i'(di — 3 ) cos 2 J 



.(23) 



we have 



„ cos a, cos a 3 r <r' / r' ,\-| 

(.5 — £) sm (a, — a 3 ) L <r \ t / J 

cos a. cos a 3 r r' / <r' \-i 

(E— S) sin (a t — a 3 ) L r \ <r / J 

cos d, cos d 3 r_ <r' . _ , / t' \ t 



(24) 



or by making 



i> = 



cos a! cos a 3 



(i2 — S) sin (a t — a 3 
cos d, cos d 3 



.#=.£ 



we have 



and making 
we have 



(E - S) sin (4, - 4,) ' G ~ F r ] 



P = D(A 3 -A 2 ) + £(A } -A 2 )^ 
Q = F(B s -B 2 ) + G(B l -B 2 ) 
p = J){a 3 — a 2 ) + E(a x — a 2 ) 

<7=P- §, 



(25) 



326 



SPHERICAL ASTRONOMY. 



(26) 



z s = C + D (a,- a 2 ) + E '(a,- a,) + F (h-b,) + G (6,-ty 
y 2 = a? 2 tan a 2 + A 2 + «a 
2 2 = x s tan d a + B 2 + 6 2 

^ = I(iJ-S)(P+^)-^(l+i?) 

rfy 2 dx % 1 .''*'» \ 

-£- = — tan a, (x 2 tan a, -f- ^, -f- a, — ^ s ) 

at ui t 

d Jl = ^ tan 4, - - (* 2 tan 6, + B^b,- z 2 ) 
at at t 

or instead of the last two, 

-jj= jj tan a 3 + — f {%* tan a, + A 3 + a 3 — y t ) 

dz 2 dx 2 1 . _» . 

— 3 = — ■ tan 3 + -, (*, tan d 3 + -#* + h - s 2 ) 

at at t 

Now although the Eqs. (26) express the values of the co-ordinates and 
components of the velocity of the body at the time of the second observa- 
tion, they involve the geocentric distances p„ p g , p„ and the radius vector 
r 2 , which are unknown, and the solution of the problem can only be ac- 
complished by successive approximations. 

First Approximation. 

Let us first neglect the terms involving aberration, and those containing as 
factors powers of <r and <r f higher than the second. This will give, Eqs. (1*7), 



F =& TT=0; TT = 



W' = 0; 



and Eqs. (18), 



«i = |-3 (y* — x * tan a 
* r 2 

by =r- —— 3 (z 2 — x 2 tan d,) 
zr 2 

a q = h = 



a 3 = — 3 {y 2 — x % tan a 3 ) 
ar' 8 

*3 = 7T~ 3 (*- - ^ taD *») 

z r 2 



(27) 



APPENDIX IX, 



327 



and as all terms involving powers of <r and <r' higher than the second are 
to be neglected we obtain from Eqs. (21), for the values of the severa. 
factors above, 

y 2 — #2 tan a, = A x " 

z 2 — x 2 tan d t = B x 

y 2 — x 2 tan a 2 = A 2 

z 2 — x 2 tan & 2 = B 2 

y 2 — x 2 tan a 3 = A z 

z 2 — x 2 tan d 3 = B 3 „ 

which substituted in Eqs. (27) give 
Apr* 



a, = 

2r 2 

a 2 = ; 



3 ' 



6i = 



^iM< 



^2 ^ 



2r, 



#3 = 



2r 2 3 ; 



h = o ; . 

* 3 = ^7"' 



and these in the first of Eqs. (26) give 

ar 2 / r' 2 <r' 2 \ 1 

** = tf+^p (2>4^ + EA { -FB Z - - #£,) .- 

2 \ <r c / r/ 

and making 

iT_ CL [DA- + ^4, -FB % -- GBJ . 



we have 






(28) 



(29) 



(30) 



(31) 



and this value of x 2l and the foregoing values of a 2 and 6 S in the second 
and third of Eqs. (26), give 

iVtan a, 



y 2 = C tan a 2 -f A 2 -f- 
z s = C tan d 8 -f # 2 -f- 



r* a 

JV tan d« 



or making 



C' = C tan a 2 + 4 a i iV 7 = iV tan a 2 ) 
C7" = (7 tan 2 + B 2 ; N" = JVtan 4 a J 



(32) 



32S SPHERICAL ASTRONOMY. 

N' 
we have y a = C -\ 3- ........ (33) 

N" 

*'=°" + 7T ( 34 ) 

and hence 

This equation must be solved tentatively. If the body be considerably 
more distant than the earth from the sun, C, C\ C" are respectively ap- 
proximations to the values of a? 8 , y 2 , z 2 5 and if considerably less distant, 

. N N' N" . . , T . 

then — , — - j and — 3- are approximations to the same quantities. It ts 
r a r 2 r 2 

evident that a value for r 2 must be selected which will make the greatest 

values of -] 3, C -\ j, and C" -\ 3-, without regard to signs 

?*2 r z r a 

r N 

less than r 2 and greater than -~> And if C ^ — 3, for instance, be the 

greatest, the value of r 2 which satisfies the equation 

will differ from the true value by a quantity less than - 2 I 1 — 1 , that 

is, less than 0.21 r 2 . 

But the solution of Eq. (35) may be greatly facilitated by means of an 
elegant geometrical construction, due to J. J. Waterston, Esq. Thus : 

Squaring the terms as indicated in the second member, recalling the re- 
lation in Eq. (7), which will give 

1 -f tan 2 a 2 + tan 2 d 2 = tan 2 2 sec 2 (3 2 ; 

substituting the values of C", C", iV 7 , N" in Eqs. (32), and making 

r = N ' . tan & 2 sec /3 S ; 
K = A 2 tan a 2 . cot & 2 cos (3 2 -f B 2 cos (3 2 ; 
(3 = K -f- C tan 6 2 sec (3 2 ; 
a 8 = ^ 2 2 + B 2 2 - K 2 ; 

equation (35) may be written 






^=^+(K+§) 8 • • • • (««J 



APPENDIX IX. 329 

(37) 



or 


a 2 \a a 4 * ri) ' ' ' 


> • 


Make 


r 2 

- = cosec 6, 
a 




then will 


r 8 a 3 

4- — 1 = cot 2 L and —. = sin 3 6 ; 

a r 2 




and Eq. (37), 


8 r 

cot 6 = - -f — r . sin 3 d ; 
a a 4 

• 








and making cot & 


= y, and sin 3 & = #, 

r 
y = -H — r a; 




Also 


y ^ ^ a 4 




or 


1 + cot 2 d = cosec 2 6 = 

sin" 






i + y 2 = 4; 

a 3 




whence 


1 

/jj — ... . _ .. 






(1 + f)~* 





(38) 



(39) 



If the curve of which this is the equation be described graphically, and the 

straight line, of which (38) is the equation, be drawn, the abscissa of their 

a 3 
point of intersection will give the value of sin 3 6, or — ; and r 2 becomes 

known. Its value may be verified by substitution in Eq. (35). 

This value of r 2 in Eq. (31) gives x 2 , and this in the second and third 
ot Eqs. (26) gives y 2 and z 2 \ also, r 2 in Eqs. (29) gives a h b { , « 3 , b 3 , and 
these in the third of (25) will give />, which with a x and b x in fourth, fifth, 

and sixth, or fourth, seventh, and eighth of (26), give — — , — 2 , and -^ t 

at a i a t 

and the values of p„ p 2 , p 3 in Eqs. (19) or (20). 

Second Approximation. 

By differentiating the equation 

r 2 = x? + yi + z 2 2 , 

and dividing by r 2 d t, we have 

drj ^x 2 dx^ yj d]h z* dz* 

dt r 2 ' dt r 2 * dt "*" r,' dt ^ ' 



330 



SPHERICAL ASTRONOMY. 



The first term of this equation becoming thus known, the values of U, W, 

U', and W, Eqs. (17), may be computed to include the third powers of 

t and <r', then the corresponding values of a x , fi„ a 2 , 6 2 , a 3 , b 3 , Eqs. (18)- 

Then, denoting by Aa„ Ab u Aa^ A& 2 , Aa 3 , Ab 3 the difference between 

the first and second values of the quantities written after the symbol A, 

and observing a like notation for the other quantities, we have for computing 

d or d ?y d 2. 

the *irst corrections to x 2 , y 2 , z 2 , — - , — - , and — - from the third and 

at at at 

foui l of Eqs. (25), 



k — Aa 2 ) ] 
h - A b 2 ) J 



Ap= D (Aa 3 — Aa 2 ) + E (Ao, — Aa 2 ) 
A ^ = i^ (A 6 3 — A b 2 ) + G (A b x 

an< ben from Eqs. (26), 
A x 2 = Ap — Aq 
A y 2 = A x 2 tan a 2 -f A a 2 

A z 2 = A #2 tan 4 2 + A b 2 

dx 2 A^9 A# 2 

A tf7 = ~ (i? " ^ " "7" (1 + B > 



dy 2 dx 2 

A -^ = A — 2 

dt dt 



tan a, (A x 2 tan a, + Aa L — Ay 2 ) 



•y £» Ci IT 

A — ^ = A -— 2 tan ^ (Aa- 2 tan ^ + A b x — Az 2 ) 

at at t J 



(41) 



(42) 



Third Approximation. 

differentiating equation (40), dividing by d t, and substituting for 

r 2 d s y 2 d 2 z 2 . . . . . . ■ . 

,-3-, -r-g , "TT> ir vames in equations (14), we nave 

dtr, _ 1 idx^ ,^l,^fl_ <^*_ M r4s x 

dt 2 r 2 \dt 2 + d* 2 **" <^ 2 rf* 2 fj ; " ,:'} I 

d 2 r 
with this value for -—? , find new values for U, W, IT, and W from Eqs. 

(17); and for a„ 6„ a 2 , b 2 , a h 6, from Eqs. (18), by including the terms 
that were omitted before. Then with the differences between these last 
values and the next preceding, form equations for the final conections by 
writing A 2 for A in equations (41) and (42). Then the final values of tbe 
required quantities become 



APPENDIX X. 3«ll 



«> 



#, +A«,+ A 2 x 2 -f &c. = Xy 

y 2 -f A 2/2 + A 2 2/2 + &o. = y, 
2 2 -{- A z 2 -f- A 2 z 2 + <fec. = z ; 



APPENDIX X. 

GEOCENTRIC MOTION. 
By the notation of the text, p. 91, 

a cos I — cos L = p . cos X, 

a sin I — sin Z- = p . sin X ; 

and by division, 

a sin I — sin L 

tan X = ; 

a cos / — cos L 

differentiating, 

d A _ (a cos I — cos L)(a cos l.dl — cos L . d L) + (a sin Z — sin Z)(a sin l.dl— &' mLdL) 
cos 2 X (a cos 2 — cos Z) 2 

_ [a 2 — a cos (Z — Z)] <^ + [1 — a cos (Z — ?)] <?Z 
(a cos Z — cos Z) 8 

But by Kepler's 3d law, 

dL : dl :: ai : 1; 

whence 

3 
c?X = a 2 . g?Z; 

which substituted above, and making 

n cos X 



a cos / — cos L ' 
gives 

rf X = P 2 . [a 9 + J - (a + J) cos (Z - I)] . dl; 

and making 

d\ — m, and d/ = n, 

we have Eq. (124) of the text. 



SPHERICAL ASTRONOMY. 



APPENDIX XI 



ON ECLIPS1J3. 

BY MK. W. S. B. WOOLHOUSE, HEAD ASSISTANT ON THE NAUTICAL ALMANAC ESTABLISHMENT. 

Eclipses, in all the varieties of aspect which they present to different places on 
the earth, form an entertaining subject for discussion; and, without considering 
the public interest generally excited by their prediction and appearance, the use of 
them, as a test of the degree of perfection of the lunar and solar tables, and in the 
determination and corroboration of geographical positions, &c, renders their accu- 
rate calculation an object of some importance. The popularity of the phenomena 
naturally called the attention of astronomers, at an early period, into the field of 
investigation, and several methods of calculation have been adopted by different 
authors at various periods. 

For the general circumstances which take place on the earth, the plan of ortho- 
graphic projection, though it can only be recognized as affording good approxima- 
tions, seems to have predominated, and to have been almost exclusively adopted 
in actual calculations. This method is explained in the astronomical treatises of 
De la Lande and Delambre, and more recently by Hallaschka, in his Elementa 
Eclipsium (Pragae, 1816), where an example is to be found at length. Various 
particulars are laid down in a more accurate manner in Memoires sur V 'Astronomic 
Pratique. Par M. J. Monteiro Da Rocha, traduits du Portugais (Paris, 1808). 

The circumstances of an eclipse for a particular place are usually calculated by 
the " Method of the Nonagesimal," which refers the bodies to the ecliptic, and an 
example of which may be seen in the work of Hallaschka above mentioned. This 
part of the subject has also been discussed analytically by Lagrange, in the Astron. 
Jahrbuch for 1782 ; and Professor Bessel has since made some important additions 
to the theory, in a paper inserted in the Astronomische Nachrichten, vol. vii., No. 
151, which is to be found translated in the Philosophical Magazine, vol. viii. 

As the numerous calculations which may be required for an eclipse, such as of 
the maps, <fcc, given in the Nautical Almanac, could not be performed without 
many perplexing references to different authors, it has been presumed that a com- 
plete and systematic set of formulae would be generally acceptable ; and such a 
conviction has led to the drawing up of the fallowing paper, which contains an 
extensive classification of useful remarks and formulae, developed and arranged 
with a careful view to their practical application, and with the endeavor to estab- 
lish a direct and uniform mode of conducting each species of calculation. 



) greatest 
) least 


61 32 
52 50 


) greatest 
) least 


9 
8 


) greatest 
J least 


16 46 
14 24 


\_ greatest 
S least 


16 18 
15 45 


) greatest 
) least 


38 35 

27 47 


l greatest 
) least 


2 33 
2 23 


i greatest 
) least 


3 4 



) greatest 5° 
\ least 4 


20 6 
57 22 



APPENDIX XI. 333 

LIMITS WHICH DETERMINE THE OCCURRENCES OF ECLIPSES. 

ELEMENTS. 

The following elements, used in the calculation of the limits, have been derived 
from the tables of Damoiseau, Burckhardt, and Carlini, viz. : 

Moon's horizontal parallax 
Sun's horizontal parallax . 
Moon's semi-diameter 
Sun's semi-diameter .... 
Moon's hourly motion in longitude . 
Sun's hourly motion in longitude 
Moon's hourly motion in latitude 
Inclination of moon's orbit with ecliptic 

LIMITS. 

For the occurrence of an eclipse of the moon : 

1. The greatest possible distance of the centres of the moon and earth's shadow 
at the time of contact, is 63' 29". 

2. At the time of true ecliptic conjunction of the moon and earth's shadow, or 
at the time of opposition or full moon, the greatest possible latitude of the moon is 
63' 45". 

3. At the time of opposition, or full moon, the greatest possible distance of the 
centre of the moon or of the earth's shadow from the ascending or descending node 
of the moon's orbit is 12° 24'. 

For the occurrence of an eclipse of the sun : 

1. The greatest possible distance of the centres of the sun and moon, at the time 
of contact, is 1° 34' 28". 

2. At the time of true conjunction of the sun and moon, the greatest possible 
latitude of the moon is 1° 34' 52". 

3. At the time of true conjunction of the sun and moon, or the time of new 
moon, the greatest possible distance of the centre of the sun or moon from one of 
the nodes of the moon's orbit is 18° 36'. 

The third of these limits applies to the true place of the node, which may differ 
considerably from the mean place. 

The most convenient and certain limits, however, will be those of the moon'a 
latitude (j8), and will be as follows : 

1. At the time of full moon an eclipse of the moon will be 
certain > ^ C < 51' 57'- 
impossible ) ^ > 63 45 

and doubtful between these limits. 



334 SPHERICAL ASTRONOMY. 

For the doubtful cases, an eclipse will result when 

a<^(P + *-<r)+*+16" 

in which P, s denote the equatorial horizontal parallax and semi-diameter of the 
moon, and *, a those of the sun. 

2. At the time of new moon an eclipse of the sun will be 

certain ' .Ixnf < <1°23'15' 



impossible ) ( > 1 34 S2 

and doubtful between these limits. 

For the doubtful eases, an eclipse will happen when 
/? <( j P_,)+ ff + s + 25' 

PARALLAX. 

If a straight line be drawn from the centre of the earth to any assumed place, it 
will be the radius of the earth for that place, and this radius we shall designate by 
the letter p. This radius p, produced upward towards the heavens, will determine 
what we shall call the central zenith, being that point which spherically deter- 
mines our true position in relation to the centre of the earth. The apparent ze- 
nith, however, is naturally determined by a line which is vertical to the observer, 
and therefore a normal to the spheroidal surface of the earth. The small angular 
deviation of this normal from the radius of the earth, or the angular distance be- 
tween the central and apparent z-niths, is what astronomers call " the angle ol 
the vertical ;" and, the earth being an oblate spheroid, it is evident that the cen~ 
tral zenith will be nearer to the equator than the apparent, and also that the hor- 
izontal parallax will always be less than that at the equator, in consequence of the 
diminution of the earth's radius in proceeding towards the poles. The effect of 
parallax on the position of a body above the horizon is to augment its zenith dis- 
tance, and for this we have the well-known relation, 

"sin par. in zen. dist. = sin liar. par. X sin app. zen. diet." 

This relation will hold strictly for the spheroidal figure of the earth, provided we 
adopt the central zenith, and that horizontal parallax which appertains to the ra- 
dius p of the place of observation. 

Consider the equatorial semi-diameter of the earth as unity, and let y denote 
the polar semi diameter, which, adopting the mean between La Lande and Delam- 

bre, will be . Let also I be the latitude of the central zenith, or what is usu- 

305 

ally called the " geocentric latitude," and I' that of the apparent zenith, which may 
be termed the spheroidal or geographical latitude. Then the co-ordinates of this 
place, referred, in the plane of its meridian, to the polar axis, will be 

g=f».sinl, y = p cos I. 

By the generating ellipse 

and therefore for the angle l\ which the normal makes with y or the tangent with 
*, we have 

iy. 1 x tan I 

~~ dx y' ' y ~~ y 3 ' 
.-. tan l = y tan J' . . . , . (t) 



APPENDIX XI. 335 

Again, the values of x and y, substituted in the above equation of the ellipse, 

give 

„ /sin 2 I , „ , \ 
P< f— — ~f cos' If = 1 ; 

and hence 

1 1 

= (2) 



i/^+^i /i + I^ 



To these may be added the following, which are sometimes useful, and directly 
deducible from the equations (1), (2), 

. y 2 tan I' (1 — e 2 ) sin V 

x = P sin I = = . . . 

v 1 + y 2 tan 2 ? VUe 2 sin 2 V 



(3) 



1 cos Z' 
y = pcosl= - z==- T= .... (4) 

Vl + y 1 tan 2 V VI - e 2 sin 2 1' 

where e= v 1 — y 2 is the eccentricity of the meridian. 
Also 



/-— — - . /l -f v* tan' 1 /' . / cos V 

The equations (1), (2) are convenient, and the latter may be simply resolved by 
logarithms, thus : 

tan xii = sin / 1/ — I 

y V v a V (6) 

p = cos ^ ) 

From (1) may also be deduced 



tan I / — ; -— 

tan x = y tan Z = = v tan I tan I' 

y 

1— y* 

tan (Z' — £) = sin 2 % 



(?) 



Here we may remark, that in reducing the geographical latitude to the geocen- 
tric with the argument I', the auxiliary arc %, being between the values of I and V , 
will be a very small quantity in defect of the argument ; and that, on the contrary, 
in reducing the geocentric to the geographical latitude, the arc x will exceed the 
argument by nearly the same quantity. Therefore, if we assume x as an argu- 
ment for the difference V — 1,0. table formed from the equation 



or 



tan (f - I) = (^-— ) sin2 Xj 

/ 1 — y 2 \ . 

/' — /=( -r — -77 ) 8m 2 v, in seconds, 

\2ytan 1"/ 



will be equally adapted to both reductions, giving nearly the mean between them ; 
and a table so constructed, with the argument x> signifying either latitude, will 
answer every necessary degree of accuracy, since the reduction itself is so small. 
In numbers we have 



y 2 609 , / 1 — y* \ 

— = ■ — , and its logarithm = 7.51641 .'.log ( ~ ) =r2.S30S4, 

y 2X304X305' & g \2ytanl"/ 



i_-y 

2 

and hence 

r - /= [2.83084] sin 2 X . 



33G SPHERICAL ASTRONOMY. 

Thus the following table has been derived: 





Difference between the Geographical and Geocentric 










Latitudes. 












Argument: ^, either Latitude. 






X 


V- 


-1 


X 


V—l 


X 


V- 


-I 


o 


o 


, 


n 





, „ 





, 


,, 


O 


9° 





O 


i5 7 5 


5 3o 


3o 60 


9 


4i 


I 


89 





24 


16 7 4 


5 5 9 


3i 5 9 


9 


58 


2 


88 





47 


17 73 


6 19 


32 58 


10 


9 


3 


87 


1 


II 


18 72 


6 38 


33 5 7 


10 


r 9 


4 


86 


1 


34 


19 71 


6 5 7 


34 56 


10 


28 


5 


85 


i 


58 


20 70 


7 i5 


35 55 


10 


37 


6 


84 


2 


21 


21 69 


7 33 


36 54 


10 


44 


7 


83 


2 


44 


22 68 


7 5i 


3 7 53 


10 


5i 


8 


82 


3 


7 


23 67 


8 7 


38 52 


10 


57 


9 


81 


3 


29 


14 66 


8 2 3 


3 9 5i 


11 


3 


IO 


80 


3 


52 


25 65 


8 39 


4o 5o 


11 


7 


ii 


79 


4 


i4 


26 64 


8 54 


41 49 


11 


11 


12 


78 


4 


36 


27 63 


9 8 


42 48 


11 


i4 


i3 


77 


4 


57 


28 62 


9 22 


43 4i 


11 


16 


i4 


76 


5 


18 


29 61 


9 M 


44 46 


11 


17 


i5 


75 


5 


3 9 


3o 60 


9 47 


45 45 


11 


17 



The difference is to be subtracted from the geographical, or added to the geo- 
centric latitude, whether it be north or south. 

It is evident from what has been said, page 334, that if Z denote the true dis- 
tance of the moon from the central zenith as it would appear at the centre of the 
earth, and Z the apparent distance from the same zenith, as seen from the place 
on the surface, where the radius of the earth is p ; and furthermore, P the equato- 
rial horizontal parallax, and z = Z' — Z, the parallax in altitude, we shall have 



sin z=/» sin P sinZ' ....... 

Substituting Z -\- z in the place of Z', and dividing by cos z, we find 



tan z 



p sin P sin Z 

1 — p sin P cos Z 



(8) 



(9) 



which are the usual formulae for the parallax in altitude. 
For the radius p of the earth we have log i/ ■ ——■ = 8.909435, and .'. by (6) 

tan $ = [8.909435] sin /, 
p = cos ip. 

The values of p so computed are given in the annexed table. 



APPENDIX XL 



337 





Log. Radius of the Earth. 






Argument: ( 


j-eocentric Latitude. 




l 


log p 


I 


log p 


I 


log p 


o 
O 


o» 00000 



3o 


9-99964 



60 


9.99893 


I 


o« 00000 


3i 


9.99962 


61 


9 


.99891 


2 


0. 00000 


32 


9.99960 


62 


9 


99889 


3 


0. 00000 


33 


9.99958 


63 


9 


.99887 


4 


9'99999 


34 


9.99955 


64 


9 


99885 


5 


9-99999 


35 


9.99953 


65 


9 


99883 


6 


9.99998 


36 


9. 9995 r 


66 


9 


99881 


7 


9.99998 


3 7 


9-99948 


67 


9 


99879 


8 


9.99997 


38 


9-99946 


68 


9 


99877 


9 


9.99997 


3 9 


9.99943 


69 


9 


99876 


IO 


9.99996 


4o 


9'9994i 


70 


9 


99874 


[i 


9.99995 


4i 


9.99938 


71 


9 


9987 s 


12 


9.99994 


42 


9.99936 


72 


9 


99871 


i3 


9.99993 


43 


9-999^4 


73 


9 


99870 


i4 


9.99992 


44 


9*999^1 


74 


9 


99868 


i5 


9.99990 


45 


9.99929 


7 5 


9 


99867 


16 


9.99989 


46 


9.99926 


76 


9 


99866 


17 


9.99988 


4 7 


9.99924 


77 


9 


99865 


18 


9.99986 


48 


9.99921 


78 


9 


99864 


*9 


9.99985 


4 9 


9.99919 


79 


9 


99863 


20 


9.99983 


5o 


9.99916 


80 


9 


99862 


21 


9.99982 


5r 


9-999 r 4 


81 


9 


99861 


22 


9.99980 


52 


9.9991 1 


82 


9 


99860 


23 


9.99978 


53 


9.99909 


83 


9 


99859 


24 


9.99976 


54 


9.99907 


84 


9 


99859 


25 


9.99974 


55 


9.99904 


85 


9 


99868 


26 


9.99973 


56 


9.99902 


86 


9 


99858 


27 


9.99971 


5 7 


9.99900 


87 


9* 


99858 


28 


9-99968 


58 


9.99897 


88 


9 


99858 


29 


9.99966 


5 9 


9.99895 


89 


9 


99857 


3o 


9-99964 


60 


9.99893 


90 


9.99857 



PHENOMENA WHICH TAKE PLACE ON THE EARTH GENERALl \ r . 



The place on the surface of the earth where the limbs of the sun and mocn first 
appear in contact will be where the penumbra first touches the earth, and, conse- 
quently, at this place the apparent contact will be in the horizon, the disk of the 
moon being wholly above the horizon, and that of the sun below it. The point of 
contact will be in the same vertical with the two centres; and, therefore, the real 
as well as the apparent places will be in the same vertical circle ; and the lower 
limb of the moon, being in the horizon, will be depressed by the whole amount of 
the horizontal parallax which belongs at that time to the latitude of the place. 
Similarly, the place which first has a central eclipse will be where the straight line 
through the centres of the sun and moon comes first in contact with the earth, a.nd 
at this place the centres of both objects will be in the horizon, that of the moon 
experiencing the whole effect of the horizontal parallax 

22 



338 



SPHERICAL ASTRONOMY. 



The same circumstances will have place where the phenomena finally quit the 
earth. 

Since the apparent places of the sun and moon are so contiguous, and the par- 
allax of the sun so small, it is evident that the relative positions will be the same 
if we give to the moon the effect of the difference of the parallaxes P — it, and 
retain the sun in his true position. This difference P — n is therefore the relative 
parallax, or that which influences the relative position of the bodies. If p be the 
radius of the earth for the place on its surface, the parallax which ought to be 
used is p (P — n). But in the following investigations, where a place is generally 
the object of determination, we cannot previously so reduce this relative parallax 
P — it. In order therefore to secure the chance of least deviation from the truth 
in this respect, we shall in these cases reduce the parallax in the first instance to 
a mean latitude of 45°, so that it will be [9.99929] (P — *). We shall conse- 
quently, to simplify the analytical expressions, hereafter denote this quantity by 
the letter P' only ; except in one or two instances, where the latitude of the place 
is known, and where it is always distinctly specified to represent the parallax 
properly reduced to that latitude, or p (P — tt). 




L Places where the different Phases are first and last seen on 
the Earth. 

Let the whole be referred to the sur- Fig. 5. 

face of a sphere concentric with the 
earth ; and let R be the relative orbit 
of the moon, which is generated by the 
differences of the motions in right as- 
cension and declination, or by the rela- 
tive motion of the moon ; iVthe north 
pole; S the sun; Sn perpendicular to 
the relative orbit, the nearest approach 
which we denote by «; C the point 
where the moon comes in conjunction in 
right ascension, and CS the difference of declination at that time, which we shall 
denote by contraction, diff. dec. Let also M M'be the positions of the moon, when 
a distance of the centres equal to A' first appears on, and finally quits the eaith ; 
MS — M'S= A, the corresponding true distance as seen from the centre of the 
earth ; ZZ the zeniths of these places on the earth, which must be respectively 
in the continuations of SM, 8 M, in order that the full effect of parallax may be 
communicated in causing the bodies to approach. 

As the apparent zenith distance of the points which experience the greatest 
effect must be 90°, we may evidently assume ZS = 90°: for contact of either 
limb of the moon with the contiguous limb of the sun, we have accurately 
•ZS — (90° — ir) + a ; for contact of either limb of the moon with the remote limb 
of the sun ZS = (90° - ») — <r; and for contact of the centres Z£=90° — *, 
By making Z S= 90°, the phase will begin with sunrise and end with sunset ; and 
it is evident that no sensible augmentation can affect the semi-diameter of the 
moon so near the horizon. The true distance 8 M of the centres being A, and P* 
the relative horizontal parallax, the apparent distance A' will be P' ~ A ; and 
by estimating positive distances from 8 towards M, in order to have the first oc- 
currence of the phase, it will be A — P' ; 

.'. A=P'+ A'. 



APPENDIX XT. 339 

Here we may notice three limiting aspects. — 

(1) When simple or exterior contact of limbs first takes place, 

A ' = s -f- or, and A = P' -p * -f* «*« 

(2) When interior contact of limbs first takes place A' = s ~ e\ when a > 9. % 

total contact first commences with A' = s — <r ; when s < <r, an annular con 
tact first commences with A' = a — s. Therefore, 
If 5 > o-, a total eclipse first begins on the earth, when 

A = P' -f * - <r. 
If s < <r, an annular eclipse first begins on the earth, when 
A = P' - s -f <r. 

(3) When contact of centres first takes place on the earth, 

A' = and A =P'. 
For the time of true conjunction in i ight ascen.-ion. assume 
D, the true declination of the moon; 
a, the true difference of right ascension, or D 'a right ascension minus ©'a 

right ascension, in space ; 
Di, the relative motion in declination, or the motion of the moon in declina- 
tion, minus that of the sun, at that time ; 
«i, the relativo motion in right ascension at the same time; 
i, the inclination of the relative orbit OR with a parallel of declination 

through the point (7, or the angle CSn; 
«, the angle under the distance and the line of nearest approach, or the angle 
MSn. This angle is always measured on the northern side of the dis- 
tance, so that when OR falls below S, or when diff. dec. C S is negative, 
it will exceed 90°. 
Then the relations of the figure will give these equations: 

•ll cos 

Hourly motion in the orbit = - — , 
J sin i ' 

arc n C=n tan «. 

For the time of describing the arc n C, or the interval between the middle of 
the general eclipse and the time of conjunction, it must be divided by the hourly 
motion in the orbit. Therefore, t denoting this interval, 



tan t = -; n = (diff. dec.) cos t ....... (1) 



(h sin t \ 



t = I — r. — I tan t. 

Assume 

,. n sin c _ _ Wain* t 1 
c=3600"X — 77 — = [3.556301 — = 

t in seconds = c tain J 

The aign of will be determined by combining the signs of diff dec. and Di ; 
and then 

time of middle = time of (5 — t . . (8) 

Also 

cos u = — (4) 

A ' 

Mn = n tan «. 



3i0 



SPHERICAL ASTRONOMY. 



Let r denote the semi-duration of the phase, or the time of describing Mn, and 
t in seconds = c tan w 
Time of 1 "^e?" , "V5 r __ ^ me Q f m [^\ e 



{+}'■!■ 



(2) 



ending 

Again, let, at the beginning, the Z NSZ = a, and for the ending, the 
Z NSZ' = b\ and, these angles being estimated from N S towards the east, wo 
shall have 

« = (-») - w, & = (-,) 4- w (6) 

and, the sun being supposed in the horizon, Z S = 90°, Z' S= 90°, 

ttrnNSZ 



cos NZ = cos NSZ sin ITS, tan ZNS = 



cos NZ = cos NSZ' sin iV£, tan Z'NS = 



cos iV^S » 
taniVSZ' 



cos 



NS 



sin / = cos a cos 6 



tan h = 



sin T = cos b cos 5 ; tan h r == — 



tan a 
sin £ 
tan 6 
sin 5 



(?) 



the latitude and hour angle I, h, relating to the first place, and I', k', to the last. 
These hour angles are measured from the sun towards the east, so that the longi- 
tudes of the places will be determined by subtracting respectively from them the 
apparent Greenwich times of beginning and ending reduced into degrees and min- 
utes, observing that positive differences will indicate east longitudes and negath»e 
differences west longitudes. 

In the preceding formulae we must use, 

r Partial ^ 
For beginning and j Total 
ending of a J Annular 



I Central 



CP'+S + a, 

Eclipse, A = ^ P>-s + ff , 



I 



Fig. 6. 



II. Rising and Setting Limits. 

The places ZZ', thus found, are the two extreme points of a series of places 
where, at the intermediate times, the same phase will appear in the horizon ; and 
for the phase of external contact of limbs, the curves which these places assume 
form one of the principal geographical limits of the general eclipse. In the an- 
nexed diagram, let M be the place of the 
moon at a time between the beginning 
and ending of the partial eclipse. Make 
Sm = A', Mm = P', and m Z= 90° ; 
then at the place Z the moon will appear 
at m, and have simple external contact 
with the sun in the horizon. The two tri- 
angles S m M, Sm' M, will give two such 
places at each instant, which, on consider- 
ing the passage of the penumbra over the 
terrestrial disk, evidently ought to be the 




APPENDIX XI. 341 

ease. Si:ice Mm = P' and Sm = A', the possibility of forming the triangles 
S m M, S m! 31, will depend on two conditions for the value of S M, viz., 
SM < Mm + Sm, 8M> Mm - Sm, or A < P' -f A' and > P' — a', that 
is, A must be between the values P' — A' and P -\- a' : this leads to two spe- 
cies of curves. 

1. When the nearest approach is greater than P' — a'. 

Here the formation of the triangles Sm M. S m' M, will always be possible du- 
ring the appearance of the phase on the earth. At the first appearance and final 
departure of the phase, S J/= Mm -f- Sm, the triangle SmM will be simply the 
line S M, and only one place Z will result. By taking positions of if on both sides 
of the middle point n, it will also appear that the relative positions of the places 
Z Z' become inverted, and that the curves described by them must intersect each 
other at some intermediate place. Hence it appears that the curve of risings and 
settings commences with a single point, which immediately after divides itself into 
two points moving in opposite directions on the earth, and which describe two 
curves intersecting each other, and finally meeting again in a single point, the whole 
forming one continued curve, returning into itself, and assuming the figure of an 
8 much distorted. At the place where they intersect, the phase will begin at sun 
rise and end at sunset, or it will begin at sunset and end at sunrise. 

2. When the nearest approach is less than P' — A '• 

In this case the triangles SmM, Sm' M, will resolve into the line S M when 
A = P' + A' and also when A = P' — A ', each of which positions will give only 
one place Z. Thus it appears that the points Z will form two distinct, oval, and 
isolated curves, the former curve being generated between the decreasing values 
A = P' + A' and A = P' — A', and the latter between the increasing values 
A =P' — A r and A = P' + A'. The leading point of the first oval and the 
terminating point of the second oval are the places where the phase begins and 
ends on the earth. The terminating point of the first oval and the leading point 
of the second oval are simply determined by using A — P' — A', and computing 
the same as for the beginning and ending of a phase on the earth. 

Let us now turn our attention to the determination of the two places Z Z' , at 
any time, or for any position of M. Join Z S and draw M d perpendicular to 
NS. 

We shall, throughout our investigation, usually denote S d by (x), dM by (y\ 
and the / dSM by S, this angle being estimated from S JV towards the east. 

To determine these quantities, let the declination of the point d~={P), which 
will a little exceed that of M, and which is distinguished from it by being placed 
within a parenthesis; then, supposing NM to be joined, the right-angled spherical 

triangle Nd M will give tan (D) = . As a is always small, the difference of 

the declinations (D) — i) = tan- 1 D maybe arranged in a small table 

cos a 

as annexed. 



342 



SPHEEICAL ASTRONOMY. 







Difference between (i>> 


and i>, 


or a corr. 














Arguments : D and a 










D 


/ 
IO 


20 


3o 


4o 


5o 


a 

6o 


70 


/ 
8o 


9° 


100 


o 
O 





o 


o 


3 


o 


„ 
o 


o 


O 


o 





I 


o 


o 


o 


O 


o 


I 


i 


i 


i 


2 


2 


o 


o 


o 


o 


I 


I 


i 


2 


2 


3 


3 


o 


o 


o 


I 


I 


2 


2 


3 


4 


5 


: 4 


o 


o 




I 


2 


2 


3 


4 


5 


6 


5 


o 


o 




I 


2 


3 


4 


5 


6 


8 


6 


o 


o 




I 


2 


3 


4 


6 


7 


9 


7 


o 


o 




2 


3 


4 


5 


7 


9 


11 


8 


o 


o 




2 


3 


4 


6 


8 


IO 


12 


9 


o 






2 


3 


5 


7 


9 


ii 


i3 


IO 


o 






2 


4 


5 


7 


IO 


12 


i5 


1 1 


o 






3 


4 


6 


8 


IO 


i3 


16 


12 


o 




2 


3 


4 


6 


9 


ii 


i4 


18 ! 


i3 


o 




2 


3 


5 


7 


9 


12 


i.5 


J 9 j 


i4 


o 




2 


3 


5 


7 


IO 


i3 


17 


20 ! 


i5 


o 




2 


3 


5 


8 


ii 


i4 


18 


22 


16 


o 




2 


4 


6 


8 


1 1 


i5 


»9 


23 


17 


o 




2 


4 


6 


9 


12 


16 


20 


24 


18 


o 




2 


4 


6 


9 


1 3 


16 


21 


26 


r 9 


o 




2 


4 


7 


IO 


i3 


17 


22 


27 


20 


o 




3 


4 


7 


IO 


i4 


18 


23 


28 


21 


o 




3 


5 


7 


T I 


i4 


£ 9 


24 


29 


22 


o 




3 


5 


8 


II 


r5 


'9 


25 


3o 


23 


o 




3 


5 


8 


I I 


i5 


20 


25 


3i 


24 


o 




3 


5 


8« 


12 


16 


21 


26 


32 


25 


o 




3 


5 


8 


12 


16 


21 


27 


33 


26 


o 




3 


6 


9 


12 


17 


22 


28 


34 


27 


o 




3 


6 


9 


13 


17 


23 


29 


35 


28 


o 




3 


6 


9 


13 


18 


23 


29 


36 


29 


o 




3 


6 


9 


i3 


i8 


24 


3o 


3 7 



The number of seconds given by this table, which we have denoted by the 
term a corr., is to be applied so as to increase D, whether it be north or south. 

The value of (D) bein^ found by so correcting D with this table, we shall evi- 
dently have 

(*) = (/)) -a, (y) = acos(D) 

(J) 



tan S=z 

(y) 






(A) 



sin S cos S' 

the quadrant in which S is to be taken being determined by (x) and (y) as co- 
ordinates. 



APPENDIX XI. 



343 



We shall afterwards have frequent occasion to use these quantities. 

If t denote the time from the middle of the general eclipse, they may be deter- 
mined more easily, though less accurate^, by means of the following formulae, 
which may readily be inferred from what has preceded. 



tan a = 



OT 



(y) = A sin 8, , 



(B) 



c 

S = ( 
(x) = A cos S, 

the upper sign being for the time t before the middle, and the under sign for the 
same time after the middle. 

Denote the / mMS by m. In the triangle mMS, which may, on account of 
its smallness, be considered as a plane one, we also have Mm=P f , Sm = a', 
and S M = a . Assume 



? = 



P' + A' 



and then 



(f -') («-f) 

P . A 



& 



As ZS, Zm maybe considered as quadrantal arcs, they will be parallel at 
the extremities S, m; and thus the / ZSM = /_ m M S = m. Therefore the 
Z N S Z= 8 ± m; and the sun being supposed in the horizon, the spherical tri- 
angle JUS Z will have Z S = 90°, and hence the places Z, Z, will depend on the 
following formulas, in which Z is called the place advancing, and Z the place fol- 
lowing. 



Place following, 
sin I = cos {8 — m) cos 6, 

Place advancing, 
sin I = cos (S + ?ri) cos &, 



tan h 



tan h 



tan 


(8- 


m) 




sin 6 


' 


tan 


(£ + 


m) 



sin «5 



(2) 



In these expressions the symbol <§ represents the declination of the sun at the 
time for which we calculate ; but for common purposes the value of <5 at the time 
of conjunction may be used in all cases. 



III. Northern and Southern Limits for any Phase. 

The determination of the extreme latitudinal limits of a phase, or of the terres 
trial lines whereon that phase will appear as the middle of the local eclipse, is the 
most complex and unmanageable of all operations which relate to a general eclipse. 
For any given phase, at different places on the earth, the moon must be so reduced 
by parallax as to touch a given concentric circle on the solar disk ; and if we con- 
sider this circle, by way of illustration, to represent, instead of the sun, the disk 
of the luminous body, the places on the earth which severally see the given phase 
must be situated in the surface of the penumbral or umbral cone, according as the 
interfering limb of the moon only approaches or projects over the centre of the 
sun; that is, the places must all be found in the intersection of this cone with tl>e 
surface of the earth. This intersection will assume a complete or partial oval 



344 



o-ffiERICAL ASTKOISOMT. 



form, according as the cone falls wholly or partially on the earth's illuminated 
disk. When it falls only partially on the earth, the extreme points will evidently 
see the sun in the horizon, and be therefore two points belonging to the horizon 
limits ; but in the other case the phase caunot at that instant be seen in the hori- 
zon. It is evident then, that these two cases have been already characterized iu 
the discussion of the rising and setting limits. Let us now suppose the bodies to 
assume consecutive positions, answering to very small intervals of time, the earth 
also turning round its axis, and we shall have a series of these ovals. It is obvious 
that the extreme geographical limits of the phase will be represented by curves 
which envelope all these ovals ; — that at each instant the place of limit, by reason 
of the compound of the motions, will be proceeding relatively in the direction oi 
the tangent to the oval ; — that there will be two of these limits when the oval 
becomes entire during the eclipse, but only one when it is always partial. This is 
the most popular and natural idea that can be formed of the nature of these limits ; 
and we may here remark, as an inference from what has been said, that if the 
rising and setting limits of any phase do not extend throughout the general partial 
eclipse, there will be both a northern and southern limit to that phase ; but that, 
on the contrary, when the rising and setting limits continue throughout the eclipse, 
there will be only one of these limits to the phase, viz. : a southern limit when the 
difference of declination at conjunction is positive, and a northern one when that 
difference is negative. 

As before, let the system be referred to a 
sphere concentric with the earth, and let M be 
the place of the moon ; Z, Z\ the zeniths of the 
places which are respectively in the northern 
and southern limits ; and m, m\ the corres- 
ponding apparent places of the moon. Draw 
the meridians N iri , NS, N m, N Z, NZ',; 
also m r, m r', and M h d h' perpendicular to 
NS; and assume Sd=(x),dM= (y),m h = x, 
h M=y, Sr = u,mr = v,Sm= A ', Zm = Z, 
Z Nm Z=M, A m NS = a\ declination of rn 

rn = D\ and the latitude of Z = I. Then the / rn NZ= k — a\m M= Z' sia Z> t 
x = m M cos M=P' sin Z cos M and y = mM sin If = P' sin Z sin M\ these by 
spherics resolve thus: 

x = P r sin Z cos M 
= P' [sin I cos D' — cos I sin D' cos (h — a') ] 

y=-P' sin Z sin M 
= P' cos I sin (h — a') 
From these we deduce 



Fig. 7. 




u = x — (x) 
— P' sin Z cos M — (a?) 
= P' [sin I cos D' — cos I sin D' cos (h 

v = {y)—y 

= {y) — P' sin Z sin M 

— (y) — P r cos l sin (h — a ') 



'')]-(*) 



(1) 



L»t us now keep our attention to the same place Z on the earth, and suppose 
the system to be in motion as in nature. The hour angle h will increase at th« 



APPENDIX XI. 345 

rate of 15° per hour, and the latitude I will by hypothesis remain unchanged; se 
that the following equations will ensue : 

- — = — P' sin 1"— - — [sin I sin D' -\- cos I cos D' cos (h — a') ] 
d t d t 

+ P' sin 1 "(l5° — p~) cos I sin D' sin (A_ a ')_^M 

= — P'sin 1"^-' cos Z+P sinl"(l5° — ^ Wn i)' sin Z sin M— ^ 
dt \ dt/ dt 

1 V = *M - P> sin 1" (l5° - £) cos J cos {h-a>) 

__ djy) __ pi gin v , ( l5 o_ ^_\ (cog ^ cos £' _ s i n Z sin D' cos M). 
dt \ dt f ' 



dt 



Now, in order that m may be the apparent place of the moon at the middle of 
the eclipse, and consequently her nearest apparent contiguity with the sun, we 
must have 

— - — = : or since w 3 + v 2 = A /2 , u — - — I- v — - = 0, which is the condition of 
dt dt ' dt 

limit. 

(L it (J ?) 
Before we substitute the preceding values of — , — , it may be observed, to 

CL z at 

avoid complexity, that the quantities P' sin i" — — , P' sin 1" - — may be neg^ 

d (x) d (y) 

lected as being very small compared with P'. 15° sin 1", -— and —^- ; also that 

Cb t (Jb t 

& may be substituted for D', which will equally serve the purpose of both northern 
and southern limits. With these modifications we have 



(2) 



-^ = P'. 15° sin 1" sin S sin Z sin M 4^ 

dt dt 

— = -~- — P 1 . 15° sin 1" (cos Z cos 5 — sin Z sin 6 cos M) j 
and, for the condition of limit, 

u \P. 15° sin 1" sin 6 sin Z sin M — ~~\ 

+ v \~jjj — P'- 15° sin 1" (cos /. cos 6 — sin Z sin S cos i¥)l =0. 

Instead of P' sin Z cos M put (#) -f- «. and for P' sin Z sin Jf put (3/) — v, and 
it becomes 

u Tl5° sin 1" (y) sin h — ^1 -f « fl5° sin 1" (») sin J -f ^1 



15° 


sin 1" 


(y) sin 


. d(r)-] 


-f « [l5 c 


sin 1" 


(x) sin ^ 


T dt 






— P' 


v 15° sin 1" 


cos Z cos <5 = 






n 

P' 

Zr- 




sin ^ - 


d(z) 

dt 

15° sin 1" 


M 


(a;) sir 


* j 


d(y) 
dt 


10 ' 15 


sin 1" 



\ cos Z : 

V cos J 



316 SPHERICAL ASTRONOMY. 

But, if ai denote the true relative motion in right ascension, and D x the true 
relative motion in declination, and D the declination of the moon, at the time oi 
true conjunction, 



d(x) 



cos Z 



dt 



= j)i 



dt 



= ai cos D ; 



£>, 



P' 



15 sin 



n v r «i cos d "l 



t» cos 6 



Make now the following assumptions 



(A) = ^?-4n = [0.58204] a t cos 3 
lo sm 1 



m 



2>i 



15 sin 1 



= [0.58204J i) 



A sin v = 
X cos v = 



(B) — (y)sin^ 

P' cos J 
(^) + (a;) sin 5 



(0) 



(3) 



P' cos-<5 

in which (A), (B) may be used as constant quantities throughout the eclipse, and 
we get 

cos Z = — ( — u sin v -f- v cos v). 



The angle r Sm is equal to the inclination of the apparent relative orbit with the 
parallel of declination; denote it by i, and then u= a' cos t', v = A' sin i, and 

.-. cosZ=A Sin(t '- -^- (4) 



which is a concise form of the condition to be fulfilled by Z and t', in order that 
the place Z may be situated in the limit of a phase. 

Since the / MS d = S, and the A M Sm = 180° — {S + i'), Z JfStoi' = 
#-{- t', we have for the triangle 31 Sm 

Mir?= A 2 + A' 2 ±2 A A' cos (£ + *'). 

Divide this by P' 2 and we get 

si,^=A!+^! ± iAA: C0S(s+0 ( „ 

for the geometrical relation between S and t', the upper sign applying to the 
northern, and the under sign to the southern limit. Add this to the square of the 
preceding equation (4), and there results 

^^ ±2 ^ cos(S+0+ ^ = 1 ... (6) 

for the determination of the angle t 1 . 

The solution of this equation is by no means very practicable ; but as a small 
error in the value of Z will not sensibly affect the angle i, we may have recourse 
to the following indirect process, in which we first consider the angle i' to be 
equal to t, which in most instances is very nearly so. The letter M designates the 
the angle Mm h. 



APPENDIX XI. 



347 



tan M 



U = A 20S i 

v = I sin t 

(D)r=2> + (« 

y = (a — a') COS (D) 

a; 



D'=6 T« 
«' = ± 



sin Z = 



cos i>' 
a') corr. 
x = {D) — D' 

x __ y 



O) 



P' cos M P' sin Jf 

the upper signs being for the northern, and the under signs for the southern limit. 

Or, if t be the time from the middle of the general eclipse, and «' the angle 
under Mm and the line of nearest approach, we shall have 



Mm sin J = n tan w = n — , and M i 



± A', 



which, observing that Mm = P' sin Z y give the following equations, wherein E 
and F are constant for all the computations. 



E = 



e (i 
uppe 
under 



± A') 
1 y sign for 



F = 



n ± A 



tan 



— t.E 



Z — 



P 
northern 
southern 

F 



\ "PP er l 
( under ) 



sign for the interval t 



£ limit. 

M=(— i)T 

. ft ' the middle. 



• (8) 



The sign of the constants i?, i* 1 , are the same as that of n ± A ' ; and when this is 
negative, the angle o>' will be in the second quadrant. 

The value of Z determined in this manner will be sufficiently approximate for 
the purposes of a general map; and where greater minuteness is wanted, it will 
serve very well to get the angle i' from the equation (4). For this we have 

cos Z 



cot t = cot v 



which may be resolved thus : 

/ cos Z 



sin <}> 



X sin v 



tan t 



tan 



(9) 



After 



2 A cos v cos 2 

is so found, which is only wanted roughly, the accuracy of the calculation 
may be tested by the equation (4) ; and then we may proceed to a correct compu- 
tation of M Z 4 by the equations (7), only using t' instead of i. We shall thus 
have in the spherical triangle ZmN, ZM=Z,Nm — %0 o — D ', and the angle 
Z m iV= M ; and I v spherics the following formulae: 

tan e = tan Z cos M 
tan (h — «') = 7j-j — jrr tan M; tan I = tan (0 -f D') cos {h — a') 



cos (0 -f- D') 
check . . 



sin Z cos M 



cos (0 -f" D') cos (h a') cos I 



(10 



For a map the equations (8) and (10) will alone be amply sufficient. In fact, 
where a very accurate calculation is wanted, the most satisfactory method will 
consist in first computing the places roughly ; then to reduce the horizontal paral- 
lax to the latitude by means of the radius p, from the table at page 337, and with 



34:8 SPHERICAL ASTRONOMY. 

the use of the value of Z, to find the augmented semi-diameter of the moon by 
means of the table at page 360, and thence the proper value of A', and then to 
follow the equations (3), (9), (4), (7), (10). 

The first and last points of these limits will have Z= 90°. For these places we 
have therefore by (5) 

P' 2 = A 2 + A' 2 ± 2 A A' cos (S+ i'). 

If we assume i = i, we shall obviously have S -f- c' == 8 -f- i = <■>, and 

A cos (S -j- t') = n, u> being the angle under the distance A and the nearest ap- 
proach n, as before used. 

.vP' ! = A 2 + A' 2 ±2 A'n, 

= A 2 — 7L* + (A'±ny. 
Consequently 

A 2 sin 2 u> = a 2 — n? = P* — (n± A ') 2 , 

which divided by A 2 cos 2 w = n?, gives 

tan w = - </P' 2 — {n± a') 2 . 

Therefore by taking the constant c used in the computation of the beginning 
and ending of a phase on the earth, we shall have 

semi-duration = c tan w = — v P' 2 — (n -+- a') 2 , 

which may be arranged for calculation as follows : 



cos w 



n ± a' 



•P 1 • , 1 

semi-duration = c — sin w , ! 



Time of j , > = tinw of middle < , >• semi-duration, 

The places of entrance and departure of the limits, by continuing the assump- 
tion i' = i, may be hence calculated as for the beginning and ending of a phase 
only using 8 ^ u instead of 5, thus : 

5^fu = I)', 1 

a =: ( — t) — u, b = ( — t) -j- w, 



For place of entrance, 



sin I = cos a cos JJ', tan /* = : — -=-, f \ XA 

sin i>' 



For /^ace of departure, 



sin £ = cos b cos 2A r tan A = 



sin D' 



Having assumed J = t, the times and places so computed will only be approxi- 
mate, though sufficiently near for general purposes. For an accurate calculation, 
we must first determine the true value of i' . Since Z = 90°, the equations (9) 
give i' = v, which is also shown by (4). We may, therefore, with the quantities 
taken out for the respective times of entrance and departure, proceed with the 
equations (C), (3), use v instead of i in (7), and then the final results will be deter- 
mined by (10). It ought, however, to be observed, that it will be advisable to 
take the time of entrance in excess to the next higher integral minute, and to re- 
ject fractions of a minute in the time of departure ; since by fixing on a time a 
trifle without the actual limits, the value of sin Z would come out greater than 



APPENDIX X ' 349 

unity, and the calculation rendered useless in consequence. The places so compu- 
ted will be accurately situated in the limiting lines, and though not strictly the 
first and last points of these lines, they will be very nearly so. 

IV. Determination of the Place where a given Phase will appear both ai 

Sunrise and Sunset. 

We have seen (page 341) that when the rising and setting lines of a phase ex- 
tend throughout the eclipse, they will compose the figure of an 8 much distorted. 
The point of intersection or nodus is a place where the phase will be seen to begin 
and end in the horizon ; that is, it will either commence at sunrise and end at 
sunset, or commence at sunset and end at sunrise. At the time of the middle oi 
the eclipse, the sun will therefore be very nearly on the meridian : if diff. dec. and 
S are of the same sign, it will be midnight, because the pole of the earth will have 
the zenith and sun on opposite sides of it ; but when those values are of different 
signs, it will be noon at the place, for then the zenith and sun will be both on the 
same side of the pole. If t denote the semi-duration of the eclipse, which begins 

and ends with the given phase, r -— will express the semi-diurnal arc of the 

sun; and.-. — tan I tan S = cos ( r — - ) = cos (r . 15°), which being nearly 

unity, we must have I ~ S or Z nearly = 90°. Consequently for the values f 

du d v 
■u, v, — , -—, at the time of the middle of the eclipse, which will be either noon 
at at l 

or midnight, we may assume sin Z = unity, and M=0 a or 180°. So we get, 

from the equations (1) and (2), page 344-5, 

« = - (*) ± P\ v = (y), 

du d{x) dv d{y) . " .... 

Tt -JT> dl == -dT ±p - 15 8inl " sm *' 

Let n denote the hourly motion on the apparent relative orbit, and i' the incli 
nation with a parallel of declination ; then 

v . du 

(t cos i' = - 
d 
or, 



ft cos i' — — , fi sin i = 

at at 



ix am i'=D 1 ) , 

m cos i = a, cos D ± [9.41796] P' sin S J * ' K ' 

d U (IT) 

The condition for the greatest phase is u f- v -— = 0, or u sin i — v cos t'= 

dt dt 

that is, 

[ — 0») ± -P'J sin i' — (y) cos t = 0. 

If t denote the interval past the time of the true conjunction, we shall have 

(x) = diff. dec. -f- 1 B v and (y) = t ^ cos I); 

.'. [ — diff. dec. ± J"] sin <' — t [Di sin t' -f- ^ cos D cos t'] = ; 

or, since D x s== ( . - \ sin t, a, cos D = | —— ) cos t, 

\ sin t / \ sin t / 

[— diff. dec. ± P'\ sin *' — t -r-^ cos (»' ~ ») = 0. 
J sm t ' 



350 SPHERICAL ASTRONOMY. 

— diff. dec. ± P' 
Assume k = j-. - . . .... (2) 

COS (t ~ l) 

k sin i' sin t . _ 

and then t = , or since ±) L = p sin t, 

X/l 

* = , or t in seconds--- [3.55630] — (3) 

I- t l . 

When difF. dec is negative, M= 180°, and the lower sign of P' must be used; 
or, as a general rule, P' must be used with the same sign as that of diff. dec, and, 
since /nearly = 90° ~ <5, we can previously correct the horizontal parallax for the 
place by reducing it to a latitude equal to the complement of 6. The value of f be- 
ing found, we shall have at the place 

when diff. dec. and <5 have -j ,.„. , , [ signs, app. time of true d =1 [ — t {-) 

which compared with the Greenwich apparent time of the true conjunction will 
show the longitude of the place. 
For the values of u and v we have 

u = ( — diff. dec. ± P') — t B v = k cos (V ~ i) — k sin i' sin t = k cos i' cos t, 
v =zt «i cos D = t D 1 cot i = k sin i' cos j. 

Let n be the nearest apparent approach of the centres ; and the semi-duration 
r will be determined by the equations 

v ri a' sin a sin i' 

cos w = — -, 



sint" A' 2>i 

and thence the latitude by the equation, 

cos (r . 15°) 
tan I = ± i - ; . 

tan 6 

Or, using the above value of v, 

k cos i A' sin w . cos (r . 15°) , m% 

tan I = ± i ' . . (5) 



A p tan $ 

the latitude being of the same name as diff. dec. 

The middle of the eclipse will not have the sun in the horizon, except k cos t = A , 
T — 0, i = 90° ~ ft, and therefore, unless these particular values should happen, 
the place will not range exactly in the line whereon the middle of the eclipse is 
seen at sunrise or sunset ; this line, which we are about to notice, will pass the 
intersection at a higher latitude, and will form a very small triangle with the 
rising and setting limits. 

V Places which will have the Middle of the Eclipse with the Sun 
in the Horizon. 

In the first place, we shall suppose the inclination of the apparent orbit to be 
the same as that of the true. The condition for the middle of the eclipse will then 
be simply to have the apparent place of the moon somewhere on the line of near- 
est approach. 

On both sides of S take Sm=-8 m! =s + <r, and m, m! will be the limits be- 
tween which the apparent place must be, in order that an eclipse may result. On 
the orbit make M 1 m' — P'. Then if m' falls between 8 and n, this will be the 
first position in which the eclipse can take place. But, if m' falls beyond the 



APPENDIX XI 



351 




point n, the first position of the moon 
will be at M t where Mn — P'; and 
in this case, for each position between 
M and M' there will evidently be a 
position of m on both sides of the or- 
bit, and consequently two correspond- 
ing places on the earth; when the m 

moon arrives at M the remote point m' will be receding from S, and will at that 
time get beyond the limit of an eclipse, so that the other point m' only will pro- 
duce an eclipse under the assigned conditions. 

Again, when m n is greater than P', it is evident that these limits will continue 
throughout the whole duration of M'M' or MM. When m« is less than P', by 
making m M" =P' the limits for an eclipse will end at the point M", and it will 
be impossible throughout the duration of M" M ". These two cases are the same 
as those distinguished in the rising and setting limits, page 340, s + a being the 
value of A '. 

To determine the times between which these phases are possible, or the semi- 
durations answering to the positions M, M', M", we shall in each instance denote 
the angle Mm n by the character a>, and the following equations will be readily 
deduced. 



(1) When n < P' — (i 

Wl = 90° 



+ *), 



+ (s + 
P' 



=C-f) 



sin a»2 



0) 



o) 2 > 90° when diff, dec. is negative. 

These semi-durations will give two times of beginning and ending; the one an* 
swering to the point M and the other to the point M ". The middle of an eclipse 
in the horizon will take place from the first beginning to the second beginning, 
and from the second ending to the first ending. 

The places will be determined by producing m M to a distance of 90° from m. 
If a great circle be drawn through 8, so as to be at this point parallel to mM, it 
will evidently intersect the former at a distance of 90° and determine the same 
place. - We shall therefore, in supposing the places to be determined in this man- 
ner, have the following formulas : 

First place of beginning, o^ = 90°, 

sin I = — sin i cos <3, tan h = -. — r f . . (2) 

sin i 

h must be taken in the 2d semicircle, or between 0° and — 180° J 
First place of ending, 
Change the name of the latitude of the place of beginning, and to the hour angle h 
apply ± 180°. The results will determine the place of ending. 



Second place of beginning, 




' 


a = — t — w 3 , 


b = - 


I + O)., 


sin / = cos a cos &, 




tan a 




sin 6 


Second place of ending, 






ein / = cos b cos 8, 


h- 


tan b 




sind 



18) 



352 SPHERICAL ASTRONOMY. 

The second places of beginning and ending will be two of the extreme points of 
the lines traced on the earth. The other two extremes may be determined by 

( e -L. a | , Ai 

computing cos « = — , and proceeding as before, observing that n must 

be considered positive, and « > 90° when diff. dec. is positive. These four extreme 
points are the same as those of the northern and southern limits, the phase being 
simply external contact. 

2) When n > P' — (s + c) and < s + a, 
The places will be determinable throughout the whole of the first ^ . . (4) 
duration found as above. 



T= \ /' 



(3) When n > s+ <r, 

n — (s + <r) 

— p, > 

n must here be considered a positive quantity, and u> will be > 90° i 
when diff. dec. is negative. 

The phase will continue throughout the whole duration, and the ex- 
treme places may be computed from this value of w according 
to the equations (3). 

Having found the limits between which the phase is possible, the places for any 
intermediate times may be determined thus, 

t denoting the time from the middle, 



-te) 



to > 90° when diff. dec. is negative, 
and the places by the equations (3). 

If n < s + o-, suppose n to be positive, and compute 

n ~ (s + a ) /c P 



-( i f)* 



cos o) = — t ; t = I I sin u. 

Then for times, without the limits of this duration, we may determine four 
places ; two with w < 90° and two with w > 90°, which will all fulfil the necessary 
conditions. 

The preceding results have been derived on the assumption of t' = i. They 
will be sufficiently approximate for a general drawing of the lines on a map, and 
more particularly as these phenomena cannot be subject to minute observation. 
When, however, from local circumstances or otherwise, greater accuracy is wanted, 
we must use the proper value of i' and the relative horizontal parallax reduced 
to the latitude thus determined. Since Z == 90°, the condition for the middle of 
the eclipse, according to the equation (4) page 346, is i' — v = or t' = v. Let the 
figure at page 344 represent the positions which answer to the particulars of the 
present case. Then as M m = M in = P', the Z Mmrri "= /_ Mm! rn. Denote 
this angle by 0; the angles Nm M, Nm'Mby M, M' ; and we shall have 

lNmS = v, i Nm' S~\W — v, 

M=0—v, Jf = 180° — v — 0, 

ZMSm — 180°-^ S— v, lMSm' = S + v, 

ZSMm=zJS + v — 9, £ SMm' —180° — (S + v + 0). 



APPENDIX XI. 



355 



With the triangles M Sm, M Sm', we hence find 



sin = —, sin (S + v) 



Sm — P' 



sin (S + v — 9) 



sin (S + v) ' 
which, for computation, may be thus arranged • 
sin (S + v) 



S m' = P 1 



sin (S + v + 6) 

sin (S + l) 



sin 9 = g . A ; 

JT 

6 to be + or — but less than 90° ; 

M =6 — v, 



9 



sin {S + v — d) 

,6 m = 



Sm' = 



sin (S + v + d) 



M'=(l8Q° — 0) — 



. (6) 



The points m, m', may in some cases be both on the same side of S, and the 
value of Sm is only necessary to indicate whether any portion of the sun id 
eclipsed or not. To have an eclipse, S m, taken as a positive quantity, must be 
less than s + o, and we must only determine a place from the angle M when the 
corresponding value of Sm is within this limit. If S m, S m', taken as positive 
quantities, are both greater than s + a, the middle of an eclipse cannot be seen 
on the earth under the assumed conditions; on the contrary, if Sm, S m' so taken 
are both less than s + «-, the angles M, M' may both be used, and consequently two 
places will be determined. In each case, similarly to (3), we adopt the formulae 

tan M ) 

sin / = cos M cos i, tan h = : — — >• ... (1) 

sin S ) 



VI. Central Line. 

The places which in succession see a central eclipse are evidently determined 
by producing S M to a distance Z from S, so that 



sin^= — 



(1) 



for then the relative parallax P' will bring the centres to a coincidence. To de- 
termine the position of the place on the earth for any given time, we have in the 
triangle N S Z, thus formed, MS = 9Q°—S, ^NSZ = S, SZ — Z, and hence 
the following formulas: 

tan = tan Z cos $, 

6 to be -f- or — and less than 90° ; 



tan h ■ 



tan S: 



tan I = tan (0 -{- $) cos k, 



cos (6 -f $) 
h to be in the same semicircle with S; 
sin 6 sin Z cos S 



check 



cos (6 -f- i) 



cos I 



(2) 



In the course of the general central eclipse, one of the places on the earth will 
have the central eclipse at noon. At this instant the bodies will obviously have 

23 



354: SPHERICAL ASTRONOMY. 

true as well as apparent conjunction in right ascension, and .\ A = diff. dec. and 
S = Q. This place is hence determined thus : 

. diff. dec. 
sin Z = — — , / = 6 -f Z, 

i fg\ 

« Z to have the same sign as diff. dec. 

App. time of true <3 = west long, of place, 

These equations (1), (2), (3), involve the horizontal parallax P', answering to a 
mean latitude of 45°, which will be sufficiently near for ordinary purposes. Where 
an accurate result is wanted, the calculation must be repeated with the use of the 
equatorial relative parallax properly reduced to the latitude thus determined. 

The first and last places on the earth which see a central eclipse, are to be 
found by the formulae at pages 338-40. 

The preceding discussions comprise all that is necessary for the calculation of 
the lines which are shown in the mops now inserted in the Nautical Almanac, and 
which are quite sufficient to indicate the general character of the eclipse that may 
be expected for any particular place. We might now proceed to show the appli- 
cation of these equations in the resolution of innumerable other curious and in- 
teresting problems; but such a field of speculation would not conform with the 
object of this paper, and may the more willingly be abandoned on the considera- 
tion that the means of solution may, in most cases, be readily elicited from the 
equations already established. The following classification of these equations will 
be found to exhibit, in a comprehensive form, all that will be requisite to direct 
and facilitate the operations of the calculator, and relieve the mind from any un- 
necessary reference or consideration. 



Notation. 

D = the J 's true declination ; 
S = the O's true declination; 
a = the true difference of right ascension in are, 

or D : s right ascension — © 's right ascension ; 
D x = the D 's relative motion in declination, 

or ]) 's motion in declination — 0's motion in declination, 
a x = the D 's relative motion in right ascension, 

or the motion of the ]> — that of the ; 
Diff. dec. = the true difference of declination at <5 in right ascension, 
viz., D 's declination — Q's declination, at that time; 
P = the D 's equatorial horizontal parallax ; 
it = the 0's equatorial horizontal parallax; 
P' = [9.99929] (P-t); 
s = the J> 's true semi-diameter ; 
9 = the Q's true semi-diameter; 
A = the true distance of the centres ; 
D\ a', s', a', the apparent values of D, a, #, A; 

m = the angle under A and n: in all cases this angle is to be taken pos- 
itively, and between 0° and 180°. 



APPENDIX XI. 355 



I. — BEGINNING AND ENDING OF A PHASE ON THE EARTH. 

1. (#, A and ai at <3 ) ; 

tan i = 1 —=, ; n = diff. dec. X cos i ; 

ai cos 1) 

i of the same sign as i), ; 

n of the same sign as diff. dec. 

2. n sin < [3 55630] 

c = j ; t = c tan t ; 

sin < to be found by combining the preceding values of cos i and tan i; 
sign of t to be determined by ditf. dec. x -Di. 

8. Time of middle = time of <3 — t ; 

f partial "j f P! + s + <r, 

„ | central ,. P', 

F0r -! total f edl P s<! ' A =/>'+„-„, 

annular J [_ P' — s -f- <r ; 

n 
cos w = — ; t = c tan w. 

A 

Time of i be f "ling [ = time of midd]e j - { 

( ending J { -f- y » ■ 

o = (— t) — «; 6 = (— «) + w. 

4. Place of beginning, (<5 at <3 ) ; 

• ; i , _ tan a 

sin / = cos a cos d : tan /* = : — - ; 

siu i 

# = apparent Greenwich time of beginning; 

longitude east =/* — H\ 

h to be in the same semicircle with a. 

6. Place of ending, (<5 at c5 ) ; 

sin I = cos b cos ■$ ; tan h = — ; ; 

8111 o 

II = apparent Greenwich time of ending; 

longitude east = h — Hi, 

h to be in the same semicircle with 6. 

6. For more accurate calculations, reduce the true relative horizontal parallax; 
oy means of the table at p. 337, to the latitudes so determined, and recompute. 



n. RISING AND SETTING LINES. 

For partial eclipse, a' = s -f- <r. 

7. Whenn> P' — a'. 

These limits will extend throughout the entire duration of the general eclipse, 
and form the listorted figure of an 8, the first and last points being the places of 
beginning and ending on the earth. 



356 SPHERICAL ASTRONOMY. 

8. When n < P' - A '. 

With P' — A', instead of a', compute as for the times of beginning at 1 ending 
on the earth ; and let these times be fa, fa. Then 

the risings j *£ n i at j P artial be g inQin S' 

in which interval the first oval will be completed : 

the settings \ e ^ y at \ 2 ' .. , -,. 

& ( end ) ( partial ending ; 

in which interval the second oval will be completed. 

The limiting places at the times fa, fa, are to be found in the same manner as 

the places of beginning and ending of a phase on the earth. 

9. Places for any times within the limits : 

Prepare the constants, p = , q = , 

and let t be the time from the middle of the general eclipse ; 

t n 

tan co = — ; A = ; 

c cos w 

w > 90° when n is — . 

10. £ = (-<) :f w . 

Use { undeJ | si S n for { IftT \ the time of middle ' 



.11. 



. m At-*) ( q ~f) 

m Y=V P^ > 



sm 
m 



to be less than 90° and positive 
L 

12. Place following, 

• 7 i a \ x 4. i, tan (£ — m) 
sin I = cos (o — m) cos o - tan h = h — s= — : 

H=. apparent Greenwich time; 

longitude east = h — H ; 

h to be in the same semicircle with 8 — m. 

13. Place advancing, 

, , tan (S 4- m) 

am I = cos (S + m ) cos o ; tan h = : — : ; 

v ' ' sin <5 

longitude east = A — H; 

h to be in the same semicircle with S -f- m. 

14. For a more accurate determination, find the values of Z>, 5, a for the given 
time, and P' s= p(P — tr) for the latitude ; thence 

(i>) = D -f (a corr. from table, p. 342) ; 
^)==(D)-<5; (2/) = aco S (/>); 

tafi>g:s (l). ^= (y > = (arJ ; 

(a;) ' sin £ cos & ' 



APPENDIX XI. 357 



P' - A 


> 


?=r 


P'+ A' 


2 


2 


sin — = 


■A 


P\ A 


-I) 



The quadrant of $ to be determined by (x), (y), as co-ordinates. 
With these values of S, m, compute the places by Nos. 12 and 13. 



HI. PLACE W*HERE THE RISING AND SETTING LIMITS INTERSECT, 
When n > P' — A '. 

15 Find P' =p (P — jt). for a latitude equal to the complement of 8 at c5» 

[i sin i' = Pi, 

H cos t f = «i cos i) ± [9.41796] P' sin 5, 

_ — diff. dec. ± P' K *ffom * sin « 

£== , £ in, seconds = |3.55G30l ^ 

cos (t ~ A« 

16. At the place, 

( the same ) • • C 12* 1 / 

W hen diff. dec. and 8 have < i- ff , [• signs, app. time of true <5 = -J h f ~~ *» 

which, compared with the Greenwich apparent time of the true c5> will determine 
khe longitude of the place. 

17. k cos t 

COS W = ; , 



A 
tan 1= ± 





L 


V srn w 






/* 


COS (r 


15°) . 





tau <5 
J to be of the same name as diff. dec. 



IV. PLACES WHERE THE MIDDLE OF THE ECLIPSE IS SEEN WITH THE 
SUN IN THE HORIZON. 

18. When n < P' — (s + ff )> compute 

cP' n± (s + e) (c P'\ . 

n = , cos u>i = -^ , r 2 = I 1 sin «# 2 ; 

n J^ \ n / 

using s + <r with a sign the same as that of n. 

These semi-durations give two times of beginning and ending ; the phenomenon 
will take place on the earth between the times of beginning and between the limes 
of ending. 

The places of first and last appearance on the earth to be determied thus : 

for first appearance, 

• 7 • • i * 7 cot * 
sin I = — : siil % cos <5, tan h = - 



sin b ' 



358 SPHERICAL ASTRONOMY. 

For last appearance, change the name of the latitude of the former place, and 
to the hour angle h apply ± 180°. 
For the extreme points compute also 

n T (* + a) / c P'\ . 
cos w 3 = -- , n = I 1 sin w 3 ; 

using s -j- a with a sigD contrary to that of n. 

Then with the values of u 2 , u> 3 , proceed as for the beginning and ending of a 
phase on the earth. 

When diff. dec. is +, j ^ I gives points meeting j ° ^j^™ \ limifc ' 
When diff. dec. is — A 2 [■ gives points meeting < , t limit 
The eclipse will be visible on both sides of the equator. 

19. When n > P' — (s + <r) and < s-{-<r, compute 

_ c P' 
n 

The phenomenon will continue throughout the whole of the duration so found. 
The two extreme points will be determined as above with the angle w 3 . 
The places of first and last appearance also as above. 

20. When n > s-{- a, compute «3, r 3, as above. 

The phenomenon will continue throughout the whole duration, and the extreme 
places will be determined by proceeding with this value of w as for the beginning 
and ending of a phase. 

These places will in this case be also those ol first and last appearance. 

21. Places for any time within the limits : 

Let t be the time from the middle, and compute 



-<#) 



If n < s + <r, this w may be taken both greater and less than 90° when t is 
greater than r 3 before found ; and then four places will be determined. In all 
other cases whatever w must be > 90° when diff. dec. is negative. 

The places to be determined by proceeding with u as for the beginning and end- 
ing of a phase. 

22. For a more accurate determination at any time : 

Find P' —p (P — tt) for the latitude before found. 
Find (x), (y), S, and A, as in No. 14. 
For the time of d form the constants 

(A) = [0.58204] ai cos D, (B) = [0.58204] D,. 

Compute v from the equations, 

(A) + (x) sin i . (B) — (y) sin * 

A cos * = ' w , A sin v = v ; p . Kif ' . 

P' cos 6 P' cos S 



9 = p , emd=g. A, 



APPENDIX XI 359 

23. Then sin (S+v) 

P ' 

9 to be + or — but less than 90°. 

24 « • , tut * t t&nM 

sin I = cos M cos a, tan h == : . 

sin 6 

If k, k', be both less than s + or, the angles M, M', may be both used in these 
equations, and two places determined. If one of the quantities k, k', be greater 
than s + <r, the corresponding M will be excluded, and only one place determined 
with the other value. If k, k, be both greater than s + a, both computations 
will be excluded, and the assumed time will be without the limits of the appear- 
ance on the earth. 

V. NORTHERN AND SOUTHERN LIMITS FOR ANY PHASE. 



(Partial } C(s + 6") + <r, 

For -| Total V appearance, A' = <(s + 6") — or, 
(Annular) ( o- — (s + 6"). 

6" is added as a mean augmentation of s. 



25. When n < P' — A ' both limits will have place. 
When n > P' — a ' only one limit will have place, vis. 

A ( Northern ) r ., , . ( — , 

A i o A-i. r hmit when nis< , ' 

( Southern ) { +. 

26. First and last points or places of entrance and departure : 

n ± a' / c P' 1 

cos w = — — , 



) • r S northern ) ,. .. 
[ sl § afor l southern f hmit 

Time of \ ^ ntrance I — time of middle 
( departure ) 



upper 
under 



Places of entrance and departure determined as in Nos, 4 and 5, for the begin- 
ning and ending of a phase, using a = ( — t) — w and b = ( — i ) + w. 

For the appearance of external contact these determinations are included in 
No. 18, and therefore need not be repeated for these limits. 

27. Places for any times within the limits : 

Prepare the following constants, using 6 at (j, 

A ' sin t 



u = a'cosi, D'=z8^U. a' = ± 



COS 



D 



Mi = — -, cos w = — =■- — as above : 

c(n ± A') P 

U PP er I sign for i nor ^ ern [limit 
under ) ° ( southern ) 






360 



SPHERICAL ASTRONOMY. 



28. Let t be the time from the middle of the general eclipse, 

cos w 



tan J = t . E, 

M 

upper ' 



sin Z 

before 



unde~r [ si S n for \ after " \ the middle ' 



29. 

tan (h — a) 



pin 



cos (0 + D 1 ) 
check . . 



taD e == tan Z cos J/j 

tan M, tan Z = tan (9 + D') cos (A — a), 

sin sin Z cos J/ 



cos (0 + D') cos (A — a') cos ^ 
< 90°, and same sign as cos 31; and k — a' to be in the same semicircle with M. 

SO. For a more accurate determination at any time, 

Find P = p (P — *) for the latitude before found. 
Also, with Z find the augmented semi-diameter s'=s + augmentation, from 

the table annexed. 



i 




Augmentation 


of the 5 *t 


5 Semi-diameter. 






I 




Argument : 


True Zenith Distance Z. 






For P 

= 54' 


Var. for 

10' 

in P. 


Z 


ForP 

= 54' 


Var. for 

10' 

in P. 


1 
z 


ForP 
= 54' 


Var. for 

10' 

in P. 


o 


r/ 


,, 





n 







n 


n 


O 


l4-0 


5. 7 : 


3o 


I2-I 


4-9 


60 


6.9 


2.9 


I 


i4-o 


5-7 ! 


3i 


12 





4-8 


61 


6-7 


2-8 


2 


i4-o 


5 '7 ! 


32 


II 


9 


4-8 


62 


6-5 


2-7 


3 


i4-o 


5-7 


33 


1 I 


7 


4-7 


63 


6-2 


2-6 


4 


i4-o 


5-7 


34 


II 


6 


4-7 


64 


6.0 


2.5 


5 


i3. 9 


5-7 


35 


I I 


5 


4-7 


65 


5.8 


2-4 


6 


13.9 


5-7 


36 


II 


3 


4-6 


66 


5-6 


2.3 


7 


13.9 


5-7 ! 


37 


II 


2 


4-6 


67 


' 5-4 


2-2 


8 


i3.8 


5-7 


38 


11 





4.5 


68 


5-2 


2-1 


9 


i3.8 


5-7 


3 9 


IO 


8 


4-4 


! 69 


4-9 


2-0 


IO 


i3.8 


5-6 ! 


4o 


IO 


7 


4-4 


! 70 


4-7 


I.9 


ii 


i3. 7 


5-6 


4i 


IO 


5 


4-3 


! 71 


4-5 


1.8 


12 


i3. 7 ; 


5-6 ! 


42 


10 


3 


4-3 


72 


4-2 


1-7 


i3 


i3.6 


5.6 ! 


43 


IO 


2 


4-2 


1 73 


4-0 


1.6 


i4 


i3.6 


5.5 ! 


44 


10 





4-i 


j 74 


3-8 


1-5 


i5 


i3.5 


5.5 : 


45 


9 


8 


4-i 


75 


3-5 


1-4 


16 


i3-4 


5.5 


46 


9 


7 


4.0 


76 


3-3 


i-3 


17 


i3-4 


5-4 


47 


9 


5 


3.9 


77 


3-i 


1 -i 


18 


i3.3 


5-4 1 


48 


9 


3 


3.9 


! 78 


2-8 


1 -i 


l 9 


l3.2 


5-4 1 


49 


9 


2 


3-8 


i 79 


2-6 


1 'i 


20 


i3.i 


5.4 ; 


5o 


9 





3.7 


80 


2.4 


1 -o 


i 21 


i3.o 


5-4 ■ 


5i 


8 


8 


3-6 


81 


2«I 


0.9 


22 


12.9 


5.3 ! 


52 


8 


6 


3.5 


i 82 


1-9 


o-8 


23 


12.8 


5-3 1 


53 


8 


4 


3.4 


! 83 


1.7 


0.7 


24 


12.7 


5-3 I 


■ 54 


8 


2 


3.3 


! 84 


1.4 


o-6 


25 


12.6 


5-2 


55 


8 





3.2 


85 


I -2 


o-5 


26 


12.5 


5-i 


56 


7 


8 


3.2 


86 


1 -o 


o-4 


| 27 


12.4 


5.! j 


57 


7 


5 


3.i 


87 


O.7 


o.3 


28 


12.3 


5.o ! 


58 


7 


3 


3-1 


88 


0-5 


0-2 


29 


12.2 


4.9 


5 9 


7 


1 


3-«) 


89 


0-3 


O-I 


U° 


12. 1 


4.9 


60 


6,9 


2.9 


90 


o-o 


O'O 



APPENDIX XI. 361 



Thjn, C Partial J rs'+ir, 

For -J Total >• phase, a' =< *'— a, 



( Annular ) 



81. For the time of o form the constants, 

(-4) = [0.58204] oi cos D, (B) = [0.58204] X> % . 

Find the values of D, 8, a, for the given time. 

(D) = D + (a com from table, page 342). 
(z)=(D)-8, (y) = a cos (JD), 

X ma _ ^ + (*) Sia * X fi in . - (^-(y ) 8in * 

A cos v = — , A sin v — — — 

Jr' cos 8 P cos 8 

32. (Z from the first computation), 



/ cos Z . tan v 

sin <p = /4/ o ^ — , tan i = 



2 A cos v ' cos 2 ^ ' 

u = A ' cos t', i)' = 5T«, 

41 

a = a' sin t', o' = ± 



cosD 

(D) = D + (a — a')corr. 
y = (a — a') cos (Z>), z = (D) — D', 

tan If = 2, sin^= * = y ■ 

a; P cos Jf P' sin Jf 

( upper ) . „ C northern ) r ., 
•} * s y siens for •< ,; [• limit. 
( under J & ( southern J 

Remaining computation the same as in No. 29. 



VI. CENTRAL LINE. 

33. The computation of the limiting times and places is comprehended under 
the head, " Beginning and Ending of a Phase on the Earth." 

34, Places for any times within the limits : 

t = the time from the middle. 

t n 

tan w = — , 



c cos u> 

u > 90° when n is negative. 



85 £ = (— »)*«; 



36 



sia Z = —, 


(8 at 6). 

tan 


sin 

tan h = — - — 


- tan S, tan 



cos (0 + 8) 



tan Z cos S, 
tan J = tan (0 -+- <5) cos h, 



check 



SPHERICAL ASTRONOMY. 
si a 9 sin Z cos S 



cos (0 + i) cos h cos I ' 
same sign as cos $, and less than 90°: 



h, same semicircle with S. 

37. For a more accurate determination at any time, find P ', S, A, as in No. 14, 
and proceed again with these as in No. 36. 

38. Place where the eclipse will be central at noon : 

(*at<5> 

. „ diff. dec. 
smZ=— — — , 1 = 6 + Z. 

Apparent Greenwich time of true 6 = longitude W. 
Z < 90° and same sign as diff. dee. 

39. For a more accurate determination, find the horizontal parallax for the lati- 
tude, and with it repeat the operation. 

[All latitudes in the preceding formulas are to be recognized as geocentric, and 
will therefore need reducing by the table at page 336.] 



Examples. 

For an elucidation of the practical application of the preceding formulae, we shall 
take the solar eclipse of May 15, 1836. At the time of new moon, viz. 2 h 7 m -o, 
the moon's latitude (3 is 25' 43', which being less than i° 23' 17'', the eclipse is 
certain. (See the limits at page 333.) The elements of this eclipse, as related to 

the equator, are 

d. h. m. s. 

Greenwich mean time of c5 hi R. A. . . . May i5 2 21 22-9 

D 's declination N. 19 25 9-8 

0's declination N. 18 57 58-8 

D 's hourly motion in R. A 3o 8 • 3 

0's hourly motion in R. A 2 28-2 

}) 's hourly motion in declination .... N. 908.7 

0's hourly motion in declination .... N. 35 -i 

D 's equatorial horizontal parallax ... 54 23.9 

0's equatorial horizontal parallax ... 8-5 

D 's true semi-diameter i4 49*5 

©'= true semi-diameter 1 5 49*9 

from which we prepare the following values : 

O , // , • 

D 's dec. . . -f- 19 25 10 D 's H. M. in R. A.. . 3o 8 
©'s dec. . . + 18 57 59 ©'s H. M. in R. A. . 2 28 

Diff. dec. . . -f 27 11 «i ..... 27 4o 

5 's H. M. in dec. . +9^9 D 's eq. hor. par. . 54 24 
©'s H. M in dec. . + 35 ©'s eq. hor. par. . 9 

Di . . + 9 24 Rel. eq. hor. par. . 54 1 5 log. 3-5i255 

const. 9-99929 

r . . . . 54 10 log. 3.5n84 



APPENDIX XI. 



363 



BEGINNING AND ENDING ON THE EARTH. 

2> x + 9' 24" .... 2.75128 (i) 
«! 27 4o . . . . 3 -22011 



J) -f 19 25. 2 

t + 19 49 

diff. dec. + 27' 11" 

n + 25 34 



cos . 
tan . 



9«53ii7 
9-97456 

9-55661 (2) 

9.97349 (3) 
3.21245 



+ i9 n 

d. h. 

i5 2 21 



56 s 

23 



sin 1 
const. 



c tan < 



3.18594 

9-53oro (2) + (3) 

3.5563o 

6-27234 (4) 



3.52106 (4) — (1) 
3.07767 



i5 2 
P' 



i 27 middle of general eclipse 

A for central phase 



54' 10" 
3o 39 



84 



= A for partial phase 



Partial. 

+ 3-18594 

3.70663 



72" 27 



( cos 4" 9*4793i 



( tan 



c . 
h. m. s. 
2 54 57 



0-49999 
3-52io6 

4-o2io5 



w 0i° 49' 



Central. 
n . -f 3.18594 

A . 3-5n84 (log./*) 

( cos -}- 9-67410 



tan 



0-27109 
3.52io6 

3 -792 1 5 



i5 2 1 27 

1 4 23 6 3o beginning 



1 5 4 56 24 ending 



d. h. m. s. 

I 43 17 

i5 2 1 27 

1 5 o 18 10 beginning 
1 5 3 44 44 ending 



(-0 

(I) 

a 
b 



— 19 49 
72 27 

— 92 16 
+ 52 38 



(-0 



o / 

— 19 46 
61 49 

— 81 38 
+ 42 o 



Place of Pautial Beginning. 



cos a . 
cos S • 

gin / . 

I . . 
Reduction 

Latitude 




tan a . 
sin . 

tan h . 

h . 
H . 



h. m s. 
-|- i-4o25i Greenwich time 23 6 3o 
-f- 9.51191 Equation . . 3 56 

1-89060 r time . . 23 10 26 

space . 347° 3r 



— 1 -89060 ( 



347 3 7 



2 10 Longitude W. 76 53 



364 



SPHERICAL ASTRONOMY. 



In the same manner may the places of partial ending and central beginning and 
ending be calculated, which will come out 



Partial ending 
Central beginning 
Central ending 



Long. E. 28 5 1 
Long. W. 98 16 
Long. E. 52 4 1 



Lat. N. 35 1 3 
Lat. K 7 58 
Lat. % 44 5o 



II. RISING AND SETTING LIMITS. 



P' . . 

S+ CT= A' . . 

P 1 — A' 
P'+A' 



54 10 




3o 39 






/ n 


23 3i 


p = 1 1 46 


84 49 


£=42 25 



Since n > P' — A', these limits will extend throughout the whole duration of 
the eclipse ; and we may therefore calculate the position of a place for any time 
between the Greenwich times i4 a 23 h 6 m 3o s , and i5 d 4 h 56 m 24 s . As an ex- 
ample, take the time i5 d o h 3o m . 









d. h. m. s. 








Assumed time 


i5 3o 








Time of middle . 

t . . . . 


i5 2 1 27 








1 3i 27 


. 3.7393^ 










c 


3-52io( 


— t 


— 19 49 

58 5i 


w . 


. . 58 5i 


I tan 


. 0.2182- 











( cos 


• 9-7 I 4o 


8 . 


— 78 40 






n . 


. 3.1859* 


m . 


34 2 


A 


. . 49 24 
. 24 42 

12 56 . 


j log A . 
( comp. 


. 3-47191 


S—m . 
S+m . 


— 112 42 

— 44 38 


h A . 
£ A — p 


. 6-52809 
. 2-88986 






q — i a 


. . 17 43 . 




. 3-02653 










Comp. log I 


>' . 6.48816 




2)18-93264 






%m . 


. 17 0-9 


sin \ m . 


. 9 -4663 2 






*n 


. 34 2 







Place Following. 



cob (S — m) —9-58648 
cos 8 . . - 1 - 9-97576 


tan (S- 

sin 6 . 

tan h . 

h . 
H . 


-m) + o.3785o 
+ 9.51191 

. _o- 86659 

. — 82° 1 5' 
8 29 


h. m. s. 
Greenwich time 3o 
Equation + 3 56 


sin I . . — 9-56224 


( time 33 56 


' / . . S. 21° 24' 

Reduction £ 


( space 8 29 



Latitude 



32 



Longitude . W 90 44 



APPENDIX XI. 



365 



Place Advancing. 



cos (S + m) 
cos 8 


. + 9. 85225 
• + 9-97576 


tan (8 + m) 
sin 8 


• —9-99444 
. + 9-51191 


sin I 


. + 9.82801 


tan h 


. + o.48253 


Reduction 


. K 42 18 
11 

. N/ 42 29 


h . . 
E . . 

Longitude . 


. — 108 i3 

. . 8 20 


Latitude . 


. W. 116 42 



By taking £ = ( — 1) + w instead of ( — t) — w, similar computations will give 
the places following and advancing for the interval t = i h 3i m 27 s after the time 
of middle, or for the Greenwich time i5 d 3 h 32 m 54 s . Much time will be saved 
by taking the computations two and two in this manner. 

HI. PLACE WHERE THE RISING AND SETTING LINES INTERSECT. 





i 


, 

90 c 

. . 18 58 






. . 71 2 




. 


P 
P — n 






9.99872 
3-51255 


p . . 


. . 54' 5" 
+ 4 36 


sin 8 . 
const. 

cosD 






3-5ii27 

+ 9.51191 

9.41796 

+ 2.44n4 
3.22011 
9.97456 




+ 26 6 


- jx COS i' 

H sin t' . 






3- 19467 




3o 42 = 


+ 3.26529 
. + 2.75128 (A) 


V 


; . i7°i' 


{ tan . 

| cos . 






. + 9-48599 
+ 9.98056 


t 


. . 19 49 


n . . 






+ 3.28473 


t' 

— diff. dec. 


ff, *;,, 2 48 
. — 27' II" 

. + 54 5 






+ 26 54 


. + 3.20790 

cos (t' ~ «) • • 9 • 99948 


fc. 1 '.. 

COB t f . 

) ■ • 1 : ■ ■ 


. + 3.20842 
. + 9-97349 

+ 3.18191 
3.26458 


sin t . 


+ 3.20842 

+ 9-53oio 

3-5563o 


A' . » 


f 6-29482 


COS w . . 

sin m >. 


9.91733 
9'75o36 


log t . . 






-f 3*01009 



366 



SPHERICAL ASTRONOMY. 



A sin o 



i5 e 
8° 4' 



tan if 
tan I 

I 
Reduction 

Latitude . 



3' 01494 

3-28473 t ' ' ' ' 

9-73021 

2-95424 A PP- time true 6 

2-68445 



9.99568 Equation. . . 
9-536i5 App. time true rf 
0.45953 



Long. 



JST. 70 5 1 
7 



time 
space 



h. m. a 

+ o 17 4 

1200 



11 42 56 at the place 



h. m. s. 

2 21 23 

3 56 



25 19 at Greenwich 



/i7'"37 8 ) 
i3 9 ° 24' ) R 



N. 70 58 



Thus we find the required place to be in longitude E. 139 24' and latitude 
N. 70 58', where simple contact will have place at sunset and again at sunrise ; 
also the middle of the eclipse would be seen at midnight if it were not intercepted 
by the opacity of the earth. The duration of the eclipse will correspond with the 
duration of the night, and therefore no portion of it will be visible. 



IV. PLACES WHERE THE MIDDLE OF THE ECLIPSE HAS THE SUN IN 

THE HORIZON. 

In the present case n is > P' — (s + a) and < s + ff . We must therefore pro- 
ceed as in No. 19. 



1. For the extreme points. 







e . 


. 3-52io6 






P' . 


. 3-5n84 


1 II 

n » . + 25 34 . 






7-03290 
. 3-18594 


— (s +■ <r) — 3o 39 




n 


. 3-84696 (1) 


— 55. 






— 2-48430 






P' . 


. 3-5n84 


w 9 . . 95 23 . 

' 

(— «)_ I9 49 -3 

*>3 95 23 


- m. 

56 
2 1 


/ cos . — 8-97246 

1 sin . . 9-99808 (2) 

39 . . . 3-845o4 (1) 
27 time of middle 


a — it5 12 
b + 5 34 


4 48 time of beginning 
3 58 6 time of ending 



APPENDIX XI. 



367 



Place of Beginning, or First Extreme Place. 

b. m. s. 
Greenwich time o 4 48 
Equation . 3 56 



cos a . 


— 9.62918 


tan 


a 


+ 0-32738 


cos S . 


+ 9.97576 


sin 


6 . 


. + 9.51191 


sin / . 


— 9-60494 


ton 


h 


. _ -8i547 


/ . . 



S. 23 45 




h 


' 
. —81 18 


Reduction 


8 
S. 23 53 


Loi 


H 

lgjitud 


. + 2 11 


Latitude 


e W. 83 29 



( time o 8 44 

Rm\ > 

( space 2 11' 



cos b . 
cos 6 . 

sin / . 



Reduction 
Latitude 



Place of Ending, or Last Extreme Place. 

h. m. 8. 
+ 9.39664 tan b • . +0.58943 Greenwich time 3 58 6 
+ 9.97576 sin & . . +9.51191 Equation . . 3 56 



+ 9.37240 


tan h 


— 1 .07752 




| time 


422 


' 
K i3 38 


h 


' 
. + 94 47 


Hin 


( space 


6o° 3i' 


5 


H 


60 3 1 









N. 1 3 43 Longitude E. 34 16 



2. For the extreme times, 

c P' 

the value of r x taken out from the preceding logarithm of is i h 57 m io*. 

n 
h. m. s. 

2 1 27 time of middle 
1 57 10 . . T X 

o 417 first appearance 

3 58 37 last appearance 



Place of First Appearance. 



sm < . 

cos I . 


+ 9«53oio 
+ 9.97576 

— 9.5o586 

1 
S. 18 42 

7 


cot l . 
sin 6 . 

tan h . 

h . 
II 


+ o.44339 
+ 9.51191 

. — 0-93148 

1 
. —83 19 

. +23 


b. m. 8. 
Greenwich time 4 17 
Equation . 3 56 


sin I . 


( time 8 i3 


Reduction 


u in \ 

( space 2 3' 



Latitude S. 18 49 Longitude W. 85 22 



Latitude K 18 49 



Place of Last Appearance. 





— 83 19 
180 


h 

H 


. + 96 4i 
. + 60 38 



h. m. s. 

Greenwich time 3 58 37 
Equation . . 3 56 



Longitude E. 36 3 



H'w\ 



time 
space 



4 2 33 
6o° 38 



368 



SPHERICAL ASTRONOMY. 



For the computation of places in this linp, we have therefore the whole range 
between the Greenwich mean times o h 4 m 17 s and 3 h 58 m 37 s . As an example, 
take the time i h 3o m . 



h. m. s. 
Time of middle 2 1 27 
1 3o 



(-0 



. o 3i 27 

o 
. — 19 49 
i5 34 

. — 35 23 
. — 4 i5 



. . 3.27577 

C J!L . 3- 84696 

n 

sin . 9.42881 



cos a 
cos 8 

sin I 



Reduction 
Latitude 



+ 9.91132 tan a 

+ 9*97576 sin <5 

+ 9-88708 tan h 



o 
K 5o 27 
11 



H 



— 9-85i4o 
+ 9'5i 191 

+ 0.33949 



— n4 35 

+ 23 29 



K 5o 38 



Longitude "W". i38 4 



h. m. a, 
Greenwich time 1 3o o 
Equation 

{time 
space 



3 


56 


1 33 


56 


23° 


29 



By similarly using the angle b, we shall find the position for the interval 3i ra 27* 
after the time of middle, or for the time 2 h 32 m 54 s ; thus, 



cos b . 

COS 8 . 


+ 9.99880 
+ 9-97576 


tan b . 
sin 8 . 


— 8-87106 
+ 9.51191 


sin I . 


+ 9-97456 


tan h . 


+ 9.35915 


Reduction 


N. 70 35 

7 


h . . 



— 167 7 
+ 39 i3 


Latitude 


K 70 42 


Longitude 


( W. 206 20 
\ E. i53 4o 



h. m. s. 
Greenwich time 2 3a 54 
Equation 3 56 

time 2 36 5o 

space 39 1 3' 



Hin \ 



The places may be computed by two together in this way ; and it will perhaps 
be a little more convenient to assume a value of t in the first instance. We may 
take any value which does not exceed r x or i h 57 m 10 s . In the present example 
we should take t= 3i m 27 s , and begin as under: 



(-<) 



' 

— 19 49 

i5 34 


— 35 23 

- 4 i5 



logt 



3.27577 
3.84696 



ime of middle . 


h. m. s, 

2 I 27 


t . . . 


3i 27 



Time before middle 1 3o o 
Time after middle 2 32 54 



and then proceed for the places as above. 



APPENDIX XI. 



369 



V. NORTHERN AND SOUTHERN LIMITS. 
I. For the partial phase, we have only southern line of simple contact. 
Constants E, cos w, Z>', a. 



s + 6" 


. . 1 4' 56" 
i5 5o 










A' . 
n . 
n— A' 


. 3o 46 
+ 25 34 

5 12 


c . 


+ 3-i8594 

— 2.49415 

— 0-69179 
3-52io6 


P' . 


— 2-494l5 
3-5ll84 






E . . 


— 7-17073 


cos w 


— 8-98231 






A' . 
cost 


3.26623 
+ 9-97349 


sin 1 


3.26623 

4- 9«53oio 






log u . 

u . 


+ 3-23972 
28' 57" 


cos D' . 


-f 2-79633 
+ 9-97448 






S . 
D 


+ 18 57-59 
+ 19 26 56 


log a' 

a' 


— 2.82185 

— ii' 4" 



The extreme places will be the same as those which have the middle of the 
eclipse with the sun in the horizon, page 366 ; and we may compute for any time 1 
between the corresponding times' of- beginning and ending, viz. : o h 4 m 48 s and 
3 h 58 m 6 s ; or we may take any value of t less than i h 56 m 39 s . For an example, 



take t 



58™ 33 s . 



Time of middle 
t 

Before middle 
After middle 



h. m. s. 
2 1 27 

o 58 33 



2 54 
o o 



Remaining calculation for the time 3h o m o 



o 

e + 5 i4-7 

D' + 19 26-9 

9 + D 1 + 24 4i*6 



+ 32 37-2 
— ti-i 

. 32 26*1 



tan£ 

COS Jf 

tan0 

sin 6 

cos . 

tan M 

tan . 

cos . 
tan (6 -f 

tan I 

I 

Reduction 
Latitude 



(-0- 

. M 



— 19 49 
100 53 

— 120 42 

+ 81 4 

+ 3o°35'-3 



t 

E 

tan «' 

cos «' 
cos w 
Z 



( sin 
( tan 



+ 9.77167 
-f 9-191 13 



. + 8-96280 


+ 8-96098 

. + 9-95835 

+ 9-00263 

-f o-8o357 


+ 9-80620 


. + 9-92.544 
') + 9-66258 


. +9.58802 


, 

. N. 21 IO-2 

7-6 



taxxZ 
sin Z . 



Comp. cos (h 
Com p. cos I . 
check * 



3-54568 

• — 7-I707.3 

. — 0.71641 

— 9.27571 

. —8-98231 

+ 9-70660 
. + 9*77167 

+ 9.70660 
+ 9. 19113 

+ 8-89773 

a') + 0-07456 

+ o.o3o35, 

-f 9-00264 



Greeuwich time 
Equation . 

time . 



m. « 
o o 
3 56 



ffin ] 



3 3 56 



N. 21 18-- 



space +45° 59' 
h . . . + 32 26 
Longitude . W, i3 33 



24 



370 



SPHERICAL ASTRONOMY. 



The calculation for the time i h 2 m 54 s is to be performed in this manner, with 
the 6ame values of tan Z, sin Z, only taking the value of M = — 120 42'. 



A MORE 


accurate Calculation for the 


Time 3 h o m o s . 




Constants 


i4% W 




flx . 
COS D . 

const. . 


+ 3«220II 

. . 9-97456 
o- 58204 


A . . 


. 4- 2-75128 

o- 58204 




3.77671 
. . +i°3 9 '4o" 


(B). , 


3-33332 


[A) . 


. +o°35'54' 



These constants may serve for the computations at all times. For the present 
-example the following is the process employed: 



D . 
a corr. . 
i . . 


+ l 9 3l 34 \(D) a • • 

f- 18 58 21 a . . 


+ 17 49-o 
+ 3-02898 


P — * . 


3-51255 


<*) • • 


f 33 i4 


cos (D) 

!°g (y) 


4- 9-97428 

+ 3- oo326 
4- 9«5i2o4 


p • • 
P' . . 
cos S 


9.99982 


Jog (a?) . 

ein S 


f 3-29973 
f 9-5i2o4 . 


3-5i237 
9-97574 




+ 2-81177 


{y) sin S 
(B) . 


4- 2-5i53o 


P' COBS 

cos Z . 
2 A cos » 


3.48811 


(x) sin $ 
{A). . 


4- o° 10' 48" 
ft 39 4o 


+ o° 5' 28'' 
+ 35 54 


9.93493 
o-6343o 


1 


+1 5o 28 


(log. 


4- 3o 26 


sin 2 <f> 
sin <t> 


9-3oo63 


(log . 
P' cos 6 


4- 3-82i38 
3.48811 . . 


4- 3«26i5o 
3.4881 1 


9«65o3a 
26°33'.i 


\ COS V • 


4- c33327 
o«3oio3 


A sin v • 
A cos v > 


4- 9-77339 
4- 0-33327 


20 . . 
COS 2 # . 


53 6-2 


a • • 


9-77843 


a A cos » 


+ o»6343o 


tan v . 


4- 9-44oi2 


. . . . 


4- 9.44012 






* . . 


4- 24 38'. 9 


( tan t' . 

J COS t 

( sin i 


4- 9-66169 
4- 9-9585i 

4. 9.62020 



• 




1 11 
i4 5o 


aug. 




12 


V 


i'5 2 


■# 


. . 


1 5 5o 


A'. 


. 


3o 52 



COS t' 


3.26764 . 

. 4- 9«9585i 
4- 3«226i5 


sin t . ■ 
cos D' 


3.26704 
r 9.62020 

2.88784 


v 

i . . 


. 4- o° 28' 3" 
. 4- 18 58 21 

. 4- IQ 26 24 


9-9745r 
2.91333 


jy . . 





APPENDIX XI. 



371 









«' . . . 


-o°i3'3 9 '- 








a . 


+ 17 49 




D . . 


. + 19 3 1 34 


(a — «' . 
(log - 


+ 3 1 28 




{a — a) corr. 3 


+ 3 - 27600 




(i>) . 


. + 19 3i 37 


. cos 


+ 9.974^8 




D' . . 


+19 26 24 


y • • 


+ 3.25028 




# 


, + 5 i3 


( tan 


+ 2.49554 
+ 0-75474 




If . . 


. + 8o° 1' 4 


(sin . 


4- 9.99338 
+ 3.5i237 
+ 3-5o5 7 5 




Z . . 


. + 33° 43 .9 


( sin 
( cos 


+ 9-74453 
+ 9'9'994 




tanZ . 


+ 9-82459 


sin Z 


+ 9-74453 




cos Jf . 


+ 9-23864 . 


. . . . 


. +9.23864 


9 . . + 6 35-9 


tan 


. 4- 9-06323 




+ 8-98317 


/>' . + 19 26.4 


sin 


. + 9»o6o34 


com p. cos (h — 


0')+ 0-09214 


q+D' 4- 26 2-3 


. cos . 


+ 9-95352 


com p. cos / 


+ o-o3i5i 






+ 9-10682 . 


check . 


+ 9.1068a 




tan M . 


- +0-75474 






' 


{ tan . 


. +9-86i56 






h — a 4- 36 i-i . 


\ 






h. m. 8. 


a — i3-7 


( cos . 


. +9-90786 


Greenwich time 3 


A . +35 47-4 


tan (0 + D') 


+ 9-68892 


Equation . 


3 56 




tan J 


. +9.59678 


( time . 
#in < 


+ 3 3 56 




Z . . 


K21 33'. 7 


( space 


+ 45° 59'. 




Reduction 


7 -7 


h . . . 


+ 35 47 -4 




Latitude 


. K 21 41 .4 


Longitude 


W. 10 1 1 «6 



This result differs materially from the former one ; but we are not to infer that 
the former position is so far wide of the truth. In general the second determination 
may be considered as an almost accurate point in the limit, and though the first 
result be some distance apart, yet it will always be very near to the limiting line, 
sufficiently near indeed for the mapping of the lines. By direct calculations of the 
eclipse for these places, the former will have an eclipse of about ^J fi of the sun's diam- 
eter, and the latter about po J„¥ °f t" ne diameter, which is too small to be perceptible. 

2. For the annular phase, we have both northern and southern limits. 
Constants E, cos w, D\ «', for northern limit. 



s + 6" . 

9 


i4' 56" 
1 5 5o 








A 
ft . . 

»4 A' . 


54 
. + 25 34 . . 
. + 26 28 . . 


. +3.i85 9 4 
. + 3-2oo85 . 




+ 3- 20085 




c 


+ 9.98509 
3 -52 106 


P' , 


+ 3-5n84 




E . 


. + 6 464o3 


cos w . 


+ 9-68901 



372 



SPHERICAL ASTRONOMY. 



+ 6o° 44 -8 



r 


. 


h. m. s. 
I 42 i4 


sin w 

n 


. + 9.94075 
. + 3-84696 

+ 3.78771 


A' 

COSl 


. + 


I.73239 
9.97349 


sin t 


1.73239 
+ 9'53oio 


log u 


. + 


1-70588 




+ 1 • 26249 


u . 


. + 


o° o'5i" 


cos D' 


+ 9'97579 


6 . 


. + 


18 5 7 59 




+ 1.28670 



D' . . + 18 57 8 



+ o° o' 19" 



The semi-duration of the northern limit on the earth is therefore i h 42 m i4 s , and 
we may calculate for any value of t not exceeding this. A calculation of the 
extreme places on the earth is to be performed the same as for the beginning and 
ending of a phase on the earth, and -will be unnecessary here. As an example, for 
a time within the limits, we shall take t = i h io m o s . 



h. m. s. 
Time of middle . 2 1 27 (— 


* • 
-*) . . — 19 49 E . 


3-62325 

+ 6-464o3 


t . . 1 10 


w' . . 5o 43 tan &>' . 

M . . — 70 32 cos o>' . 
M . . + 3o 54 cos w . 


+ o- 08728 


Before middle . 5i 27 . . 
After middle . 3 1 1 27 . . 


+ 9-80147 
+ 9-68901 




Z. . + 5o3..3Jf n t ' 
( tan Z . 


+ 9-88754 
+ 0-08423 


Remaining calculation for the timt 


s 3 h u m 27 s . 




tanZ . 
cos M . 


+ o-o8423 sin Z. 
+ 9.93352 cos M . 


+ 9-88754 
+ 9-93352 


$ + 46 io< 2 tan 
2>' + 18 57-i sin d 
9 + D' + 65 7.3 . cos . . 


+ 0-01775 

+ 9«858i7 comp. cos (A — a') 

+ 9-62397 comp. cos / . 


+ 9-82106 

+ O- l5622 

+ 0-25693 


tan M . 


■f 0-23420 . check 

. + 9-77706 


+ 0-23421 


' ( tan . . 
A — a! +45 44-6. J 
«' . + 0-3 1 cos . . 


+ 0-01126 


b. m. s. 

e 3 i 1 27 

3 56 


+ 9-84378 Greenwich tin 
') + o-33374 Equation . 

. + 0-17752 | time 

. K56° 23'- 8 Hm ( spact 
10 -4 h . . . 


h . + 45 44-9 

tan I 


. 3 f5 25 


Reduction 


1 + 48° 5i' 
+ 45 45 


Latitude 


. N". 56 34 •- Longitude 


W. 3 6 



The calculation for o h 5i ra 27 s is to be performed in the same manner, with 
Jf = — 7 o°32'. 



APPENDIX XI. 



373 



A MORE ACCURATE CALCULATION FOR THE TlME 3 h II* 

o / II I II 

+ 19 33 27 j_ ,™ a . . +23 6 



27 s . 



s . . 


1 11 

. 14 5o 


aug. . 


. . 9 


«' . . 

r 


. 14 5 9 
i5 5o 


A . 


5i 



i . . 


+ *8 58 28 


a 


4- 3.I4I76 


P-* . 


3-5i255 


<*) • • 


4- 35 


cos (D) 

lo g (y) 


+ 9-974I9 

4- 3-ii595 
4- 9«5i2o8 


p - • 

P' - . 

cos S 


9.99901 


log(z) . 
sin <5 


4- 3*32222 

-f 9.51208 . 


3.5n56 

9-97574 




4- 2»8343o 
+ o° II' 23" 

4-1 39 4o 


(y) sin & 
(B) . 


4- 2-62803 


P' cos S 

cos Z . 
2 X cos v. 


3.48730 


(x) sin 8- 
(A). . 


+ o° 7' 5" 
4- 35 54 


9'8o33i 
o- 63740 


f 


4- 1 5i 3 


(log. 


4- 28 49 

4- 3-23779 
3-4873o 


sin 2 <p 
sin i> 


9-16591 


(log • 
P' cos & 


4- 3-82367 
3-4873o . 


9.58296 

22°3o'-4 


\ COS V • 


4- o- 33637 
o-3oio3 


X sin v • 
X cos v • 


4- 9- 75o49 
4- o-33637 


2 f . 

COS 2 <P . 


45 0-8 


2 


9-84939 


i A cos v 


4- o- 6374o 


tan v . 


4- 9-4i4i2 




4- 9-4Ui2 






*' . . 


4- 2o°9'.4 . 


/ tan t' 
J cos t' . 


. 4- 9-56473 
4- 9.97255 






s 




( sin t' 


4- 9-53728 



. 


1.70757 




1-70757 


COS l' 


4- 9.97255 


sin 1 . 


. 4- 9-53728 




4- 1. 6801 2 


cos D' 


4- 1-24485 


u , . . 


4- o° o'48" 


- + 9*97577 


6 . . . 


4- 18 58 28 


a 


4- 1 • 26908 


D' . . . 


4- 18 57 4o 


. 4- o° 0' 19" 






a . 


. 4- 23 6 


i) . . . 


4- 19 33 27 


( a — 0' . 


. 4- 22 47 


(a — a') corr. 


1 


(log - 

. COS 


. 4- 3-r3577 


(J>) . . 


4- 19 33 28 


• + 9-974i9 


D' . . . 


4- 18 57 4o 
4- 35 48 


y • • 


4- 3-10996 
. 4- 3-33ao3 


if . . . 


4- 3c°5 7 '-i 


( tan 

( COS 


• "+ 9'7779 3 
. 4- 9-93328 






P' . 


4- 3.39875 
. 4-3-5ii56 


2 . . . 


4- 5o° 27'. 9 


j sin 
( cos 


. 4- 9-88719 
. 4- 9-8o383 



374 



SPHERICAL ASTRONOMY. 



O ' 

9. . +46 5-8 
J}' , + 18 57-7 
B + D> +65 3-5 


tan Z . 
cos M . 

tan 
sin 9 

. cos . 

tan M . 
1 tan . 

( cos . 
tan (9 + D' 
tan Z 

/ . . 

Reduction 

Latitude 


+ o-o8336 
+ 9-93328 

+ o-oi664 
. + 9.85764 
. + 9.62499 

+ o> 23265 

• + 9' 77793 

+ o-oio58 


sin Z ... + 9-88719 
cos M . . . + 9-93328 

+ 9.82047 
comp. cos (h — a) + o-i5588 
comp. cos I . -f o-2563q 


O ' 

h — a + 45 4i-9 - 
a . + o-4 

h . + 45 42-3 


. check + o- 23265 


. + 9.84412 

) + Q-33249 

+ 0-17661 

. If. 56° 20'. 5 
10 .4 


Greenwich time 3 11 27 

Equation . 3 56 

( time . + 3 i5 23 




U 1 space + 48° 5o'-8 
h . . . . + 45 42 -3 




N. 56 3o .9 


Longitude . W. 3 8 .5 



VI. CENTRAL LINE. 
We have, at page 363, found the semi-duration of the central appearance on the 
earth to be i h 43 m 17 8 , which is therefore the greatest value of t for this phase. 
As an example for a time within the limits, take the same value of / as in the two 
preceding examples. 



b. in. s. 
Time of middle . 2 1 27 



Before middle 
After middle 



o 5i 27 
3 11 27 



(-0 

u 

. s 

. 8 




t 

c 
tan u) 

C03 u» 

n . 

A . 
P' . 



Z 

tanZ 



Z. . + 4,35.6 \Z 

Remaining computation for the time 3 h 1 i m ■27 s . 

tan Z . . + 0-06994 sin Z. . 

cos S . . + 9.92905 cos S 

+ 9-99899 

+ 9.84898 comp. cos h 

+ 9-64339 comp. cos I 

check 



9 + 44 56-o 


tan 6 


& + 18 58. 


sin 9 


e + S + 63 54-o 


. cos . 




tan # 


' 
. + 44 56-6 . 


( tan k 




„ cos h 




tan (9 



+ o- 20559 

+ 9-79354 
+ 9.99913 



. 3-62325 

. 3-52io6 

. 0-10219 

. 9.79246 

. 3.18594 

. 3-3 9 348 

. 3-5n84 

. 9-88164 

. 0-06994 

+ 9-88164 
+ 9-92905 

+ 9-81069 
+ o- 1 5009 
+ c- 24480 

+ o-2o558 



ft). 



+ 9.84991 
+ 0-30990 



h. m. s. 
Greenwich time 3 11 27 
Equation . . 3 56 



tan I 

I 

Reduction 
Latitude 



+ 0-!D( 



. K55° 18'. 7 
10 »6 



K 55 29- 



#in J 

h . . . 

Longitude 



time 
space 



. 3 i5 23 

+ 48° 5i' 
+ 44 5 7 
W. 3 54 



APPENDIX XL 



375 



D . 

* corr 

i 
<*) - 

P — * 

• 

P' . 



4- 19 33 27 

1 
4- 18 58 28 



A MORE ACCURATE CALCULATION. 
. + 23 6 . . 



(JD) a 



o 35 o 

3.5i255 
9-999 03 
3-5n58 



4- 3i°52'. 7 



cos B 

(y) • 

j tan 5 

I cos S 



+ 3.I4I76 
+ 9-974I9 

+ 3. 1 1595 

+ 3*32222 

4- 9-79 3 73 
4- 9-92899 

-f- 3-39323 
3-5n58 



49° 35'- 6 j 



e + s 



+ 44 55-8 
+ 18 58»5 
+ 63 54-3 



4-44 5 7 .5 



tanZ 

cos # 

tan 
sin e 
cos . 

tan $ 
tan h 

cos A 
tan (9 
tan / 



Reduction 
Latitude 



4- 0-06994 
+ 9 9 28 99 
+ 9-99893 
4- 9-848 9 5 
+ 9-6433r 
-f o-2o564 
+ 9'79 3 73 
+ 9'999 3 7 

4- 9.84980 
+ o- 3 1000 
+ o- 15980 

K 55° 18'. 7 



sin Z . 

tanZ . 

sin Z . 

cos S . 



N. 55 29 -3 
Central Eclipse at Noon. 



com p. cos h 
com p. cos / 
check . 



Greenwich time 
Equation . 

L time 
Hin 

I space . 

h . . . . 

Longitude . 



+ 9.88165 
4- 0*06994 
+ 9.88165 
+ 9-9 28 99 
4- 9-81064 
4- o. i5o2o 
4- 0.24480 
4- o-2o564 





3 


56 


4- 3 


i5 


23 


4-48 c 
4-44 


5o' 

5 7 


• 8 
•5 



W. 3 53 -3 



Diff. dec. 
P . 
sin Z 

Z . 
S . . 
I . . 
Reduction 
Latitude 



3.21245 
3-5ii84 
9.70061 




Time of 6 
Equation 

Long, i 



time 



space 



2 25 19 
36° 20' 



W. 



N. 49 17 

By assuming a series of times, and so computing, in conformity with the preced- 
ing examples, a series of points on each of the several limits will be determined; 
and these points being laid down in a geographical map, with respect to latitude 
and longitude, it will be easy to trace the lines through them. In this manner has 
the following map been executed, the assumed law of projection being that the 
parallels of latitude are concentric and equidistant circle?. This projection will 
be found very suitable when an eclipse, as in the present instance, extends com- 
pletely round one of the poles of the earth. In other cases, any hypothesis what- 
ever may be assumed, with respect to the law of projection, provided the geo- 
graphical sketching and eclipse-lines be both laid down on the same principle. 
(See Fig. 11.) 



PRINCIPAL LINE^ FOR THE SOLAR ECLIPSE OF MAY 14-15, 1836 

Fig. 11. 




APPENDIX XI. 377 

PHENOMENA FOR A PARTICULAR PLACE. 

I. — Eclipses of the Sun. 

The chief objects of determination for any particular place are — 

1. For a partial eclipse, its magnitude, and the times of beginning, greatest 
phase, and ending. 

2. For a total eclipse, the times of external and internal contact of limbs, or the 
times of partial and total beginning and ending. 

3. For an annular eclipse, the times of exterior and interior contact of limbs, or 
the times of partial and annular beginning and ending. 

Also, to secure certainty in the observation, it is necessary to determine, in each 
case, the particular points on the limb of the sun, as related either to the vertical 
or a circle of declination, where these contacts take place ; and hence the general 
configuration of the eclipse. 

We first proceed to find expressions for calculating, at any time, the apparent 
relative position of the two bodies, and the augmentation of the semi-diameter of 
the moon. The parallax in altitude depends on the Eq. (8) or (9), page 336. It 
will here be necessary to investigate the effects which this parallax will produce 
in the right ascension and declination of the moon. These might be accurately 
determined by the theory of the small variations of spherical triangles, but not 
quite so simply as in the following manner: — Assume, as before, 

/, the geocentric latitude of the place ; 
it. A., the true right ascension of the moon; 

D, the true declination of the moon, + north, — south ; 
h, the true hour angle of the moon, + west, — east ; 
r, the distance of the centres of the earth and moon. 

Then if, from the earth's centre, we take 

a, on the intersection of the planes of the meridian and equator, -4- towards 

upper meridian ; 
y, in the plane of the equator, + west, — east ; 
z, parallel to the earth's axis, + north, — south ; 

we shall have, for the position of the moon, 

x = r cos D cos h, y = r cos D sin h, z = r sin D; 

and, for the position of the observer, 

(x) = ft cos I, (y) = 0, (z) = p sin /. 

Thus the position of the moon, in relation to the observer as an origin, will be 

x = x — (x) = r cos D cos h — p cos I ; 
y' =y — {y) z=z r cos D sin It • 
z' = z — (z) = r sin D — p sin I ; 

and hence, D', b! denoting the apparent declination and hour angle, and r' th« 
distance of the moon from the observer, we shall have 

x = r' cos D' cos K ==r cos D cos h — p cos I ; 
y' = r' cos D' sin ti = r cos D sin h ; 
z' = r' sin D' = r sin D — p sin I. 



375 



SPHERICAL ASTROKOMY 



Therefore, as cot K' = — , tan D' = - sin A', - = sin P, we find 
2/ V r 



p sin P cos I 
cos D sin A 

/ p sin P sin /\ sin fe' 

\ sin B / sin A 

, / p sin P 



cot A' = cot A 



tan 



tani) 



/ p sin /* \ 

cot A — cot A = I ■ — ; I 

\cos D sin A/ 



tan P tan D' 



sin A' 



/ p sin P \ . 

= ( fi - r— j;) sm ' 

\cos i> sin A/ 



.(1) 



which present a direct method of calculating the apparent position of the moon, 
at any time, from that of the true. The former of these equations is evidently 
subservient to the other, and must necessarily be computed first. As the calcula- 
tion of these expressions will, in general, require seven places of figures, it will be 
more convenient to determine the simple effects of the parallax, or the small dif- 
ferences A.R. — A.R.', D — D', for which other expressions may be derived from 
them. Let A.R. — A.R! = A' — A = A A, and D — D : = a D ; then by multi 
plying the equation 

p sin P cos / 

cot h — cot A = — - — - 

cos D sin h 



by sin h sin A', the left-hand member will become sin (A' — A) or sin A A. 



Again we have 



But 



nSlfl? cos / . 

. ' . sin Aft- =r sin A . 

cos D 



tan D tan D' p sin P sin I 



__/> 



sin A sin A' cos D sin A' 



tan D tan D' tan D — tan D' / 1 1 



sin A sin A' 



+ ( J_ _ i ) tmjy . 

\sin A sm A / 



sin (D — D') sin A' — sin A _. 

; rr + tan D' 



A cos D cos D' sin A sin A' 



sm 



sin A D 2 sin £ A A cos (ft + | A A) _ 

t ^ 7v + • — r~- — n tan D ■ 

A cos i> cos 1) sin A sin A' 



. , psmP sin £ , _ - 

Equate this with - — : — -, and we nnd 

n cos 1) sm A 

sin A B p sin P sin £ 2 sin -J A A cos (A + -£ A A) sin 2X 



cos P cos i)' 



cos D 



sin A' 



' cos D' 



_ . , sin A A p sin P cos £ sin A' 

But 2 sin £ i A = -— — - = — . — — - . 

cos ^ A A cos J) cos ^ A A 



APPENDIX XI. 379 

Substitute this value and multiply by cos B cos B' and we deduce 

T> T • 7 TV 7 • TV C0S ( k + i A ; *)1 

sin A P = a sin P I sin / cos D' — cos / sin D' — --=—, — - I 

K L cos i A A J 

TVe shall therefore have, for the parallax of the hour angle, and that of the decli- 
nation, 



(p cos t) sin P . ,. 

sin A h = — — sin A' 

cos i> 



A P = sin P I sin I) cos P' — (p cos £) sin P' — r — ' I 

L cos | A A J J 



(2) 



These are still however not adapted for direct calculation, since they involve 
the apparent quantities A', B', which it is our object to determine. The only use 
that can be made of them is, first to use the true quantities, in order to get the 
parallaxes and apparent values approximately, and then to repeat the operation. 
To avoid this difficulty, substitute in the former A + A A instead of A', and in the 
latter put P — A P instead of P', and we get, by expansion, 



, p cos I sin P . . _ r . . . ,. 

A h = — (sin A cos A A -\- cos A sin A A) ; 



sin A P = p sin P cos A P I sin I cos D — cos I sin P — - I 

L cos i A A J 

4- p sin P sin A P I sin I sin P -f cos I cos P — — '- I. 

L cos £ a A J 

Divide these by cos A A, cos A P, respectively, and solve for tan A A and tan A P, 
and we find 



tan A A = 



(p cos I sin P\ . , 
r - -— 1 sin A 
cos P / 



tan A P = 



(p cos / sin P\ 
— I cos A 
cos P / 

• -d T • 7 r. 7 • T, cos (A + ^ A A)"| 

• sin P I sin I cos B — cos / sin P — 7-— -, — ■ I 

L cos & A A J 



(3) 



1 • d F • ; • n . 7 ■ n cos (A + i A A)l 

1-psinP Ism/ sin P -4- cos I cos P 5 — — - I 

L cos £ A A J 

. . . |~ tanP cos(A + iAA)l 

(p sin / sin P) cos P | 1 . i — -— - — - I 

vp ' L tan £ cos £ A A J 

cos (A -f- \ A A)l 



1— (psinZsinP) sin PI H , 

vp ; L tan /tan D 



cos i A A 



(4) 



These expressions are all of them perfectly rigorous, and better suited to calcu- 
lation than they would appear at first sight. The process of the calculation, in 
which five places of figures will be sufficient, is more detailed in the following 
equations : 



__ (p cos /) sin P 
cos B 



tan A A = 



n sin A 



1 — n cos A 



(5) 



380 



SPHERICAL ASTRONOMY. 



c = (p sin l) sin P ; 
n x = & tan D ; 



* = co8(A + » A -ff.coW 



cos ■£ A A 



n 2 



tan 2? 



_ c cos D (1 — ni) 

tan aD = - r-7—7— . 

1 — c sin i> (1 -f- n 2 ) 

The expression (4) for tan a D may, however, be neatly resolved by means 
spherical triangle as follows : 

Assume Fi §- 9. 

cos (A -f I A A) 



(•) 



of a 



cos (A) = 



cos -J- A A 



• (a) 



(A) being very nearly equal to A + £ A A And let A 7 " be the 
north pole, J£ the central zenith, and M the moon ; then N M 
= 90° - D, NZ = 90° - I, and the / A' = A. Without 
changing these values of NM, NZ, let us suppose the hour 
angle A 7 " to become increased to the value of (A); and "with the 
triangle so constituted suppose the altitude of the moon to be / 

e, so that ZM=- 90° — t\ then the spherical relations \, 

sin Z M cos M = cos NZ sin NM — sin NZ cos N M cos A 7 ", 
cos ZM= cos A'Z cos A'JLT-f sni ^^ sin A\3f cos A 7 ! 




will give 



cos t cos M = sin J cos 2> — cos / sin D cos (A) 

• 7 n 7 • y. cos (A -f -J a A) 
= sin / cos V — cos I sin 2> — - — (• 

cos { aA 

sin e = sin / sin 2> + cos ^ cos -0 cos (A) 



sin £ sin D -f- cos Z cos D 



cos (/i -f- -J A A) 



cos |aA 

Comparing these with the former expression of (4), we have therefore 

(p sin P) cos £ 
tan a JD= —^-7 — : ' . . cos Jf ... 
1 — (p sin jP) sm £ 

Before this can be used the angles M and e must be determined. 
Draw ZD perpendicular to MN, and by spherics, 

tan ND = tan NZ cos N 

sin M D tan M = tan Z D = sin A 7 " D tan N; 

sin A'Z' 
. • . tan M = - — — — - tan N 



tan MZ 



MD 
tsmMD 



-, or cot MZ = cot MD cos M 



Also by (c) 



sin ND 
sin Jf 2? 



tan M 

tan A 7 "' 



cos A 7 " sin Jf 
cos Jf' sin N 



cos A 7 " sin NZ 



cos if sin Jf Z 



(/) 

(6) 
, (c) 



APPENDIX XI. 



381 



Let now NB = d, and MB = MN— Q = 90° 
(a), (6), (c), (d), {e), (/), will give the following: 



cos (h) = 



cos (A -+- i A A) 



tan 6 = cot £ cos (h) . . . 

tan M == , m tan (A) 

cos (0 -f Z>) v 

tan « = tan (0 -{- D) cos if . 

sin cos (A) cos I 



cos (0 -f- B) cos J/ cos £ 



(0 -f- B) ; and the equations 
(«)" 

./:.:. (6) 

w 

(d) r 

00 



.(?) 



_ (p sm P) cos e 

tan A B = - . ' . — cos if. 

1 — (p sin P) sin e 



(/) 



in which the equation (e) is used as a check on the preceding computations. This 
check affords a good security to the accuracy of the work, and gives to these equa- 
tions a decided preference over those of (6), although a trifle more perhaps in point 
of calculation. They have also another advantage, inasmuch as if may be consid- 
ered as the parallactic angle, and « the altitude of the moon ; the former of these 
is useful in determining the position of the line joining the centres of the two bod- 
ies in relation to the vertical, and the other is useful in finding the augmentation 
of the moon's semi-diameter, which we shall now consider. 

If s' denote the moon's apparent semi-diameter, and s her true semi-diameter as 
seen from the centre of the earth, the actual semi-diameter of the moon will be 
represented by both r sin s, and r' sin s' ; also, if a perpendicular be drawn from 
the centre of the moon upon the radius p produced, this perpendicular will be rep- 
resented by both r sin Z, and r sin " 



We must therefore have - — — = 



sin s sin Z 

Let M be the true position of the moon, in the preceding figure, and sin Z M 
sin / NZM= sin NM sin N will be sin Z sin /_ NZM '== cos D sin h; for the 
apparent position of the moon the angle NZM will remain the same, and sin Z 1 
sin / NZ M = cos D' sin A'. 

sin Z cos D' sin h' 

' ' sin Z cos D ' sin h ' 



Also, by means of the equations (8) and (9), page 336, 



sin Z' p sin P sin Z' 

sin Z p sin P sin Z 



p sin P sin Z p sin P sin Z 



tan z = 



1 — p sin P cos Z 



sin s 
sin x 



Z' cos B' 



cos B 



sin h! 
sin h 



1 — p sin P eos Z 



.(8) 



All the preceding formulae are strict in theory. It now remains to consider 
what allowances may be made and what facilities given in their actual calculation. 
In the first place the value of cos £ A h may be safely assumed equal to unity, 
and may therefore be rejected in the equations (2), (4), (6), and (7), so that (h) = 
h ■ -f- i A h ; it may be shown that this supposition cannot involve an error of 
more than 0".03 in the value of A B. 



382 



SPHERICAL ASTRONOMY 



Also, as the arcs P, A h, A D, are small, we must have very nearly 



sin P 



= sin 1" 



[4.6S557], 



tan A h tan A D 



A A 



aD 



tan 1" = [4.68557], 



•where P, A h, A i), denote respectively the numbers of seconds they contain. 
These equations may be made more exact, for the limits between which the angles 
are always comprised, by adopting numbers differing a little from sin 1" and 
tan 1"; thus, by assuming 



sin P 



= [4.68555], 



tan A h 



= [4.68561]. 



The first supposition will not in any case involve an error exceeding that of 
0''.05 in the value of P, nor the second an error of more than 0".l in the value of 
A h, and these are much too small to merit attention ; the latter assumption ap- 
plies equally the same to A D. 

Thus we shall have (h) = h + $ A h, sin P= [4.68555] P, A h = [5.31439] 
tan A h, A D = [5.31439] tan A D ; also, A h = A a, the parallax in right as 
cension. The equations (3) and (7) may therefore be commodiously arranged as 
follows : 



A = cP; 

n = k cos h; 



c = [4.68555] p ; 
m = A cos I ; 

k sin 



~^1 ' 

cos D r • . • 



A a = [5.31439] 



1 — 71 



(9) 



By taking h less than 180°, positively or negatively, A a will have the same 
sign as h. 

tan 6 = cos (h) cot I ; G = cos (h) cos I 

sin 9 



tan M = 



cos (0 + D) 
B = cos M cos e, 

ti = A sin e : 



tan (/*) ; tan t = tan (9 -(- D) cos M 

sin # 



check 



cos (9 + D) 



AD= [5.31439] ■ 

1 Ui 



(10) 



The auxiliary arc 9 may be taken out in the first quadrant, -{- or — ; calling 0° to 
180° the first semicircle, and 180° to 360° or 0° to — 180° the second semicircle, 
the parallactic angle M must be taken out in the same semicircle with h; and 
A D will have the same sign as cos 3f. 

It will appear by the preceding investigations that the values of A a, A J), so 
deduced, are the quantities to be subtracted from the true values of A.M., D, to 
get the apparent. 

As the number n is always very small, the values of com p. log. (1 — n) to the 
fifth place of figures may be comprised in the following useful Table under the 
title of Correction of Log. Parallax, and conveniently taken out with the nearest 
third figure of the argument. 



APPENDIX XI. 



383 







Correction of Log. Parallax. 












Argument 


log. n. 








Log n 


Corr. 


Log n 


Corr. 


Log n 


Corr. 


T 

Log n 


1 
Corr. L 


3gn 


Corr. 


5-oo 





7. 100 


54 


7»4oo 


109 


7.700 


218 8 


000 


436 


• IO 







no 


55 1 




4io 


112 




710 


223 


010 


447 


.20 


1 




120 


5 7 




420 


n4 




720 


229 


020 


457 


• 3o 


1 




i3o 


58 




43o 


117 




73o 


234 


o3o 


468 


.40 


1 




i4o 


60 




44o 


120 




740 


240 


o4o 


479 


• 5o 


1 




i5o 


61 




45o 


123 




75o 


245 


o5o 


490 


.60 


2 




160 


63 




46o 


125 ! 


760 


a5i 


060 


5oi 


.70 


2 




170 


64 




470 


128 




770 


257 


070 


5i3 


.80 


2 




180 


66 




480 


i3i 




780 


263 


080 


525 


.90 


3 




190 


68 




490 


1 34 




790 


269 


090 


53 7 


6- 00 


4 




200 


69 




5oo 


i3.7 




800 


275 


100 


55o 


.10 


6 




210 


71 




5io 


i4i 




810 


281 


no 


563 


.20 


7 




220 


72 




520 


1 44 




820 


288 


1 20 


5 7 6 


• 3o 


9 




23o 


74 




53o 


1 48 




83o 


294 


i3o 


590 


• 4o 


11 




240 


76 




54o 


i.5 1 




84o 


3o2 


i4o 


6o4 


• 5o 


i4 




25o 


77 




55o 


i55 




85o 


3o8 


i5o 


618 


.60 


17 




260 


79 




5 60 


1 58 




860 . 


3i5 


160 


632 


.70 


22 




270 


81 




5 7 o 


162 




870 


323 


170 


647 


.80 


27 




280 


83 




58o 


1 65 




880 


33i 


180 


663 


'9° 


34 




290 


85 




590 


169 




890 


338 


190 


678 


7 «oo 


43 




3oo 


87 




600 


i 7 3 




900 


346 


200 


694 


7-000 


43 




3io 


89 




610 


177 




910 


355 


210 


710 


•OIO 


44 




320 


9 1 




620 


181 




920 


363 


.220 


727 


.020 


46 




33o 


9 3 




63o 


186 




93o 


3 7 i 


• 23o 


744 


• o3o 


4i 




34o 


95 




64o 


191 




940 


3 79 


• 240 


761 


• o4o 


48 




35o 


98 




65o 


i 9 5 




95o 


388 


• 25o 


779 


• o5o 


49 




36o 


100 




660 


199 




• 960 


398 8 


• 260 


798 


• 060 


5o 




370 


102 




670 


204 




.970 


407 






• 070 


5 1 




38o 


104 




680 


209 




.980 


417 






.080 


52 




390 


107 




690 


213 


7 


'99° 


427 






•090 


53 


7 


4oo 


109 


7 


• 700 


218 


8 


• 000 


436 1 






7-100 


54 












| 






This 


correction is ac 


ditive when n 


is positive, and 


subtracts 


e wh< 


;n n is 


negativ< 


;. For the part 


illax in declina 


tion it will alw 


ays be adc 


itive 


if the 


moon bt 


j above the hori 


zon. 








' 



For the augmentation of the moon's semi-diameter we may assume cos z ss 1 and 
Z = 90° — e, so that 

!_' — 1 _ 1 

5 1 — p sin i 3 sin e 1 — Wi ' 
Wi being the number which enters into the computation of A D. Hence 



,___s_ __ [ 9.4353 7] P 
1 — ni 1 — tii 



(ID 



384 SPHERICAL ASTRONOMY. 

This and the last formulae for A a, A D, entirely preclude the necessity of having 
recourse to a table of the sines and tangents of small arcs, and possess much uni- 
formity and simplicity in their application. 

To get the relative parallax of the moon with respect to the sun, we must use 
P — 7r, instead of P. If, therefore, P' denote the value of p (P — tt), or the rela- 
tive horizontal parallax reduced to the latitude of the place, we must use sin P', 
instead of p sin P, in the preceding formulae. 

The determination of the apparent relative positions of the centres of the two 
bodies, as well as the augmentation of the semi-diameter of the moon, at any time, 
has now been reduced to a practical and expeditious 6et of formulae. A series of 
these apparent positions of the moon, with respect to that of the sun, will trace 
out her apparent relative orbit; and the contact of limbs will evidently take place 
when the apparent distance of the centres becomes equal to the sum or difference 
of the semi-diameter of the sun and the augmented semi-diameter of the moon. 
For a distance equal to the sum of these semi-diameters we shall have partial be- 
ginning or ending ; for a distance equal to their difference we shall have 



total 
annular 



beginning or ending, when s' •] \_ [> ff . 



Since the hour angle of the bodies is subject to the rapid variation of nearly 15° 
per hour, the effect produced by parallax will be of so irregular a nature as to 
give a decided curvature to the apparent relative orbit of the moon. This curva- 
ture will be more strongly characterize* 1 when the eclipse takes place at some 
distance from the meridian or near to the horizon ; and the apparent relative 
hourly motion of the moon, even during the short interval of the duration of the 
eclipse, will, through the same irregular influence, experience considerable varia- 
tion. These circumstances will, in some measure, vitiate any results deduced in 
the usual manner, by supposing the portion of the orbit described during the 
eclipse to be a straight line, and using the relative motion at the time of apparent 
conjunction as a uniform quantity. The method we are about to pursue is very 
simple, and consists in assuming any time within the eclipse, and computing for 
this time the relative positions and motion of the bodies, and thence finding, with- 
out any reference whatever, either to the time of the middle of the eclipse or to 
the time of conjunction, the times of beginning, greatest phase, and ending, and 
the relative positions of the bodies at these times. The nearer the assumed time 
is to the time of the greatest phase, the more accurately will the time of that 
phase be determined ; and, similarly, the nearer that time is to the time of begin- 
ning or ending, the more certainty will attach to the determination. 

To find the apparent relative motion of the moon, we must first determine the 
variation which takes place in the parallax. For this, take the equations (2), p. 
879, viz.: 

. sin P' cos I . . . 

sin A a = sin A h = ~ — sin h, 

cos D 

sin A D = sin P' I sin I cos D' — cos I sin D' = ; — I : 

L cos t A h J 

or, substituting small arcs instead of their sines, 

cos I . . . 



Aa=F 



cos D 



cos (h + $ AhY 



A D = P' I sin I cos D' — eos I sin D' 

L cos | A h 



APPENDIX XI. 385 

Since a portion of the apparent disk of the moon is projected on that of the sun, 
the apparent declination D' can differ very little from <5. As the hourly variations 
of these small quantities are only required approximately, we may therefore use 
6 instead of D' and neglect A h, so as to have 

n> C0S l • i 

A a = P — sin h, 

cos I) 

A Z> = P' (sin I cos S — cos I sin <5 cos h) ; 

which expressions, though rough values of A a, A D, will give their hourly varia- 
tions pretty accurately. For these, observing that h is the only quantity which, 
by its rapid variation, has any sensible influence on these values, we have by 
differentiation, 

d(A a) / dh . V cos I 



( n dh . A cost . 

F — sin 1") -cos A, 
dt / cos 1) 



dt 

— : = I P -r- sin 1' I cos I sin o sin h. 

dt \ dt / 

But by the equations (9), 



Substitute, therefore, 



m = [4.68555] P' cos/, 

n = [4.68^55] P' -^— cos h. 
J cos D 



™ cos I r • 

P' tt cos h = [5.31445] n, 

cos D L J ' 



and we 



P' cos 1 = [5.31445] m; 

d(A a) /dh . „\ 

-^ = [5.31445] (-smr) W , 

d(AD) rp , _ /dk . „\ 

— -7T — = [5.31445] I — - sin 1 1 m sm S sin h. 

If we adopt 14° 29' as a mean value of — , we shall have — sin 1" = [9.40274] 

and [5.31445] (— sin 1") = [4.71719] or [4.7172]. Therefore, if (5), the value 

of the sun's declination at the time of the middle of the eclipse, be adopted in tne 

,d(AD) 
value of — - , we may form the constants, 

Qi = [«172], ) 

& = [4.7172] m sin (S) J {ll) 

and then, using a «i, A A in place of y *\ -^ — \ W e shall have 

Qjt <i t 



Aa 1 = Q 1 n, ) 
ADt—Qzsinhf ^ C ' 



which offer a simple calculation. 
25 




386 SPHERICAL ASTRONOMY. 

Let now, at any assumed time within the Fig. 10. 

duration of the eclipse, S and M be the ap- 
parent positions of the centres of the sun 
and moon; and B M E an arc of a great cir- 
cle coinciding with the relative direction of 
the moon's motion at that time, which arc 
we shall first adopt in place of the curvilin- 
ear orbit actually described. On the circle 
of declination S N, demit the great circle 

perpendicular M d, and suppose B and E to be the positions of the moon at the 
respective times of partial beginning and ending of the eclipse, and ?* the middle 
point. Assume SB = S E — s' -f o- = A', Sd=x, dM = y, SM= W Sn = n, 
Z JSTSM — S, Z BMd — Z dS n = i, and the Z B S n = /ESn = *>, Also, 
for simplicity, let x h y x denote the hourly variations of x and y. 

In determining the value of x we shall require the a correction, which will 
reduce the declination of the point M to that of d. This correction is shown in a 
table at p. 342 ; but, as this small correction may be wanted more accurately 
than can be obtained from that table, we shall here give some factors for its de- 
termination, from which, in fact, the table alluded to has been derived. The cor- 
rection will resolve as follows: 

, . tan D 
tan (B) = ; 



.-. tan [(B) — D] 



tan B 

tan D ..'„/, x 

cos a _ tan I) (1 — cos a) 

tan 2 B cos a -f- tan 2 B 



Or, supposing cos o = 1 in the denominator, 

Suppose, now, a to be expressed numerically in minutes, and (B) — Bin see* 
onds; then 

tan [(B) — D] = [(B) — B] sin 1" j 

sin - = | sin 1' = (30 sin 1") a. 

Therefore, by substitution, we find 

(B) — B = (900 sin 1" sin 2 D) a\ 
Consequently, assuming 

F= 90000 sin 1" sin 2 B = [9.63982] sin 2 D t 
vre shall have 

a corr. z=(B) — B = F. (^\ . 

The value of F, argument B, is contained in the follow ing small table 






APPENDIX XT, 



387 



Factor F for « correction. 


D 


.P 


D 


F 


D 


F 

















O 


• ooo 


IO 


•i4 9 


20 


.280 


I 


.oi5 


II 


.164 


21 


.292 


2 


• o3o 


12 


.178 


22 


• 3o3 


3 


• o46 


i3 


• 191 


23 


• 3i4 


4 


• 061 


i4 


•2o5 


24 


.324 


5 


.076 


i5 


.218 


25 


•334 


6 


• 091 


16 


•23 1 


26 


•344 


7 


• i<>6 


17 


.244 


27 


• 353 


8 


• 120 


18 


• 256 


28 


• 362 


9 


• i35 


19 


.268 


29 


.370 


IO 


•149 


20 


.280 








<z corr. in second* = F . ( — 1 
\*Q/ 




a denoting the number of minutes it contains. 



From what has preceded, it is evident that a = a — a «, is the apparent dif- 
ference of the right uscen-ions of the bodies, and that D = D — A B is the appa- 
rent declination of the moon ; and that 



x == [D' -f (« — A «) corr.] — i ) 
y = [« — A «] cos D' ) 

and consequently also 

#1 = A — A D, J 

D'J* ' 



(14) 



(15) 



3/1 = ( <n A «i) COS 

Moreover, the figure occupying so small a portion of the sphere, and being com- 
posed of arcs of great circles, we may, without any appreciable error, treat these 
arcs as straight lines ; thence we shall obviously have 



tan S- 



_V_ 
cot « = — , 



W: 



9 



sin S cos S 



Hourly motion in the orbit 



(1«) 



n=Wcos(S + t), 

Again, in the triangles B S M, E S M, 

£ BSM=z» + {S + t ), 
wid. consequently, by plane trigonometry, 

B M= -2L sin [a, -f (S + «) ], 

COS ta J 



^ESM^o — iS+t), 



J5M/= JLsin [« — (S+i)l 



« if =TF sin <£ + •). 



388 SPHERIJAL ASTRONOMY. 

With the above hourly motion in the orbit we shall therefore have 

B M = TFe ° S ' sin [» + (8+ 1) I 

3/1 COS 0) L x • ' 



Time of describing 



v W cos t . . 

n M= sin (S + t), 



EM= sin [« - (8+ t) 1 

?/l cos W ' J 



Let, now, ti, U, be corrections to be applied to the time assumed to get the times 
of beginning and ending, and (t) the correction for the time of the greatest phase. 
Then we have evidently 



C t x \ (BM\ ( negative J 

< (t) > = the time of describing < n M \ with a < negative > 
( t 2 ) {EM) (positive ) 



To have these times expressed in seconds, assume 

IF cos 1 „ „ IF cos 1 [3.55l__ 

c = X 3600" = . ^ ± (17) 

1/1 cos a 1/1 cos o) ' 

and then we shall derive 

tx = c sin [— (S -f — «], U = c sin [— (& + 1) -f «], 

(t) = c cos w ain [ — (8 4- ], 
and hence 

C beginning } C c sin [ — (S + «) — w] J 

The time of 1 greatest phase > — assumed time -\- •< c cos w sin [ — (8 + 1) H (18) 
( ending ) ( c sin [— (# + 1) -f w] J 

It has been observed, that any one of these values will be the more to be de- 
pended on the more nearly it approximates to the assumed time. Thus, if the 
assumed time be within ten minutes or so of the end of the eclipse, the point M 
will approximate so closely to the point E, that no sensible error can arise by 
supposing the small portion ME of the orbit to be a straight line, and to be 
passed over by the moon with a uniform motion. This circumstance renders it 
advisable, in the first instance, to take the assumed time near to the time of the 
middle of the eclipse, so as to give a good result for the time of the greatest phase, 
and results for the times of beginning and ending, which may be nearly equally 
relied on. Such a computation will be sufficiently exact for the usual purposes of 
prediction. When the time of beginning or ending is wanted to great minute- 
ness to compare with observation, it will only be necessary to repeat the operation 
for a time assumed as near as convenient to the first determination, which will 
mostly give within a fractional part of a second of the true theoretical result ; a 
degree of accuracy, however, seldom wished for, and quite unsupported by the 
present state of the lunar theory. 

To fix on a time near to the middle of the eclipse for the radical computation, 
one of the most simple expedients will be to determine roughly the time of th« 
apparent conjunction. 



APPENDIX XI. 389 

We shall now briefly consider the apparent positions of the moon, as related to 
Lhe sun's centre. 

It is clear that S is the angle of position of the moon's centre from the north 
towards the east, at the time assumed ; also that the angle 2V S B = o> -{- ' is the 
similar angle of position from the north towards the west at the time of begin- 
ning; and that the angle N S E = w — <■ is the angle of position from the north 
towards the east at the time of ending ; and that the angle N S n = t is the same 
angle towards the west at the time of the greatest phase. Therefore, by estima- 
ting all these angles towards the east we shall have 



( beginning ~\ C( — 0~~ w ) 

At ^greatest phase > Z of D ' s centre from N. towards E. = ■?(— \ 

( ending ) ((-*) + «•>) 



(19) 



In the computation of the parallax in declination, we find an angle M, which in 
practice may be supposed to be the angle N S Z fur the assumed time, the zenith 
Z being reckoned towards the east ; consequently, at this time we shall have 8 — ' M 
for the angle of position of the moon's centre from the zenith towards the east. 
At any other time the parallactic angle M for the latitude of Greenwich may be 
taken from the following table, arguments the corresponding apparent time and 
the sun's declination. This table, for any other place, may be computed by for- 
mulae, such as at page 381, viz. : 

tan = cot I cos h, tan M = ■ — -— - tan h. 

cos (0 -f b) 

h being the angle answering to the apparent time. 

Those who may be engaged in the computation of eclipses, for any particular 
places, will find considerable facility in the formation of similar tables. 



For an occultation of a star by the moon, the argument, instead of the apparent 
time, will be the star's hour angle, or the sidereal time minus the star's right as- 
cension. In this case the required positions will be those of the star with respect 
to the moon's centre, which will therefore be different from the angles of position 
for a solar eclipse, in which the moon's centre is referred to that of the sun. The 
angular positions of the contacts at immersion and emersion will consequently be 
determined in the same way as for an eclipse of the sun, and will be estimated in 
the opposite directions. Thus, for an occultution, 

^Srs?}^*'^.^.^i:=.{[iss:=:j+:f 

And so must 180 c be applied to the other angles of position, as expressed for a 
solar eclipse: this will make the expressions for the direct images of occupations 
the same as those for the inverted images of eclipses of the sun, in estimating the 
contacts either from the north point or from the vertex 



390 



SPHERIUAL ASTRONOMY, 







Parallactic Angles for the Latitude of Greenwich, 






(same sign as h) 




Arguments : Apparent Hour Angle and Declination. 






Hour Angle h. 


Dec. 
North. 






































o 

















































IO 


20 


3o 


40 


5o 


60 


70 


80 


00 


100 


1 10 


120 


i3o 


140 




o 


o 





o 


■ 
































o 


O 


8 


i5 


22 


27 


3i 


35 


37 


38 


39 


38 


3 7 


35 


3i 


27 


i 


o 


8 


i5 


22 


27 


32 


35 


3 7 


38 


3 9 


38 


3 7 


34 


3i 


27 


2 


o 


8 


16 


22 


28 


32 


35 


3 7 


38 


39 


38 


37 


34 


3i 


27 


3 


o 


8 


i'6 


22 


28 


32 


35 


37 


38 


3 9 


38 


36 


34 


3i 


26 


4 


o 


8 


16 


23 


28 


32 


35 


3 7 


38 


39 


38 


36 


34 


3i 


26 


5 


o 


9 


16 


23 


28 


33 


36 


38 


3 9 


39 


38 


36 


34 


3o 


26 


6 


o 


9 


17 


23 


29 


33 


36 


38 


39 


3 9 


38 


36 


34 


3o 


26 


7 


o 


9 


'7 


24 


29 


33 


36 


38 


3 9 


39 


38 


36 


34 


3o 


26 


8 


o 


9 


17 


24 


29 


34 


36 


38 


39 


39 


38 


36 


33 


3o 


25 


9 


o 


9 


17 


24 


3o 


34 


37 


38 


39 


3 9 


38 


36 


33 


3o 


25 


IO 


o 


9 


18 


25 


3o 


34 


3 7 


39 


39 


3 9 


38 


36 


33 


3o 


25 


ii 


o 


9 


18 


25 


3i 


35 


37 


39 


39 


3 9 


38 


36 


33 


29 


25 


12 


o 


IO 


18 


25 


3i 


35 


38 


3 9 


40 


3 9 


38 


36 


33 


29 


25 


i3 


o 


10 


'9 


26 


3i 


35 


38 


39 


40 


39 


38 


36 


33 


29 


25 


i4 


o 


IO 


'9 


26 


32 


36 


38 


4o 


40 


3 9 


38 


36 


33 


29 


25 


i5 


o 


IO 


l 9 


27 


32 


36 


39 4o 


4o 


3 9 


38 


36 


33 


29 


24 


16 


o 


j i 


20 


27 


32 


3? 


39 


4o 


4o 


4o 


38 


36 


33 


29 


24 


J7 


o 


ii 


20 


28 


33 


37 


39 


4o 


4i 


4o 


38 


36 


33 


29 


24 


18 


o 


1 1 


21 


28 


34 


38 


40 


4i 


4i 


4o 


38 


36 


33 


29 


24 


l 9 


o 


j i 


21 


29 


34 


38 


4o 


4i 


4i 


4o 


38 


36 


33 


29 


24 


20 


o 


12 


22 


29 


35 


39 


4i 


4i 


4? 


4o 


38 


36 


33 


29 


24 


21 


o 


12 


22 


3o 


36 


39 


4i 


42 


42 


4o 


39 


36 


33 


29 


24 


22 


o 


12 


23 


3o 


36 


4<> 


42 


42 


42 


4i 


39 


36 


33 


29 


24 


23 


o 


i3 


2 3 3i 


37 


40 


42 


43 


42 


4i 


3 9 


36 


33 


29 


24 


24 


o 


;3 


24 32 


38 


4i 


43 


43 


42 


4i 


39 


36 


33 


29 


24 


25 


o 


i4 


25 33 


38 


42 


43 


43 


43 


4i 


3 9 


36 


33 


29 


24 


26 


o 


i4 


26 34 


3 9 


42 


44 


44 


43 


42 


3 9 


36 


33 


29 


24 


27 


o 


i4 


26 35 


4o 


43 


44 


44 


43 


42 


39 


36 


33 


29 


24 


28 


o 


i5 


27 35 


4i 


43 


45 


45 


44 


42 


4o 


3 7 


33 


29 


24 


=9 


o 


if 


28 36 


4i 


44 


45 


45 


44 


42 


4o 


3 7 


33 


29 


24 



By subtracting the parallactic angle, for the respective times of beginning, 
greatest phase, and ending, from the foregoing angles of position of the moon's 
centre from the north towards the eas% wo shall evidently obtain the same angles 
from the zenith or vertex towards the east. 

If, however, the operation be repeated for the accurate determination of the 
times of beginning and ending, we shall have in the calculations the angle M also 
at these times. Let «i, ta u M x be the angles appertaining to the beginning, and 
» 2 , a>2. Mo those for the ending, and we shall evidently have the following values, 
which will be more accurate than the preceding : 



APPENDIX XI. 



391 







Parallactic Angl 


2s for the Latitude of Greenwich 
















{same sign as h) 




Arguments 


: Apparent Hour Angle and Dedication. 










Hour Angle h. 


Dec. 

South. 













o 


o 


o 


o 





































10 


20 


3o 


4o 

o 


5o 



60 




70 




80 



90 




100 




no 

9 


120 




i3o 



140 




o 





o 





o 


O 


O 


8 


i5 


22 


27 


3i 


35 


37 


38 


39 


38 


37 


35 


3i 


27 


I 


o 


8 


i5 


21 


27 


3i 


34 


3 7 


38 


3 9 


38 


37 


35 


32 


27 


2 


o 


8 


i5 


21 


27 


3i 


34 


3 7 


38 


39 


38 


37 


35 


32 


28 


3 


o 


8 


i5 


21 


26 


3i 


34 


36 


38 


39 


38 


37 


35 


32 


28 


4 


o 


7 


i5 


21 


26 


3i 


34 


36 


38 


3 9 


38 


37 


35 


32 


28 


5 


o 


7 


i5 


21 


26 


3o 


34 


36 


38 


39 


39 


38 


36 


33 


28 


6 


o 


7 


M 


20 


26 


3o 


34 


36 


38 


3 9 


3 9 


38 


36 


33 


29 


7 


o 


7 


'4 


20 


26 


3o 


34 


36 


38 


39 


39 


38 


36 


33 


29 


8 


o 


7 


i4 


20 


25 


3o 


33 


36 


38 


3 9 


39 


38 


36 


34 


29 


9 


o 


7 


i4 


20 


25 


3o 


33 


36 


38 


39 


3 9 


38 


37 


34 


3o 


IO 


o 


7 


i4 


20 


25 


3o 


33 


36 


38 


39 


39 


3 9 


37 


34 


3o 


ii 


o 


7 


i4 


20 


2.5 


29 


33 


36 


38 


3 9 


39 


39 


37 


35 


3i 


12 


o 


7 


i4 


20 


25 


29 


33 


36 


38 


3 9 


4o 


39 


38 


35 


3i 


i3 


o 


7 


r4 


T 9 


25 


29 


33 


36 


38 


39 


4o 


39 


38 


35 


3r 


i4 


o 


7 


i3 


J 9 


25 


29 


33 


36 


38 


3 9 


40 


4o 


38 


36 


32 


i5 


o 


7 


i3 


l 9 


M 


29 


33 


36 


38 


39 


4o 


4o 


39 


36 


32 


16 


o 


7 


i3 


l 9 


24 


29 


33 


36 


38 


4o 


4o 


4o 


39 


37 


32 


17 


o 


7 


i3 


r 9 


24 


29 


33 


36 


38 


4o 


4i 


4o 


39 


37 


33 


18 


o 


7 


i3 


'9 


24 


29 


33 


36 


38 


4o 


4i 


4i 


4o 


38 


34 


T 9 


o 


7 


i3 


J 9 


24 


29 


33 


36 


38 


4o 


4i 


4i 


40 


38 


34 


20 


o 


7 


i3 


z 9 


24 


29 


33 


36 


38 


4o 


4l ': 4l 


4r 


3 9 


35 


21 


o 


6 


i3 


l 9 


24 


29 


33 


36 


39 


4o 


41 


42 


4i 


39 


36 


22 


o 


6 


i3 


■i9 


24 


29 


33 


36 


3 9 


4i 


41 


42 


42 


4o 


36 


23 


o 


6 


il 


18 


24 


29 


33 


36 


39 


4i 


42 


43 


42 


4o 


37 


24 


o 


6 


i3 


18 


24 


29 


33 


36 


3 9 


4i 


42 


43 


43 


4i 


38 


25 


o 


6 


i3 


18 


24 


29 


33 


36 


39 


4i 


43 


43 


43 


42 


38 


2.6 


o 


6 


i3 


18 


24 


29 


i33 


36 


39 


42 


43 


44 


44 


41 


39 


27 


o 


6 


i3 


18 


24 


29 


33 


36 


39 


42 


43 


44 


44 


43 


4o 


28 


o 


6 


12 


18 


24 


29 


33 


3 7 


4o 


42 


44 


45 


45 


43 


4i 


29 


o 


6 


12 


18 


24 


29 


33 


37 


4o 


42 


44 


45 


45 


44 


4i 



For 



( beginning 
■s greatest phase 
( ending 



(—1)- 

Z of D 's centre from K towards E. = ^ ( — 1 ) 

.(-!.) + 



Z of ) 's centre from vertex towards E. 



](-*)-M I (20) 

((- i 2 ) + « a — J/J 



These angles relate to the natural appearance or direct images of the bodies. 
For the same angles, as they will appear through ar. inverting telescope, ± 1S0 C 
must be applied : this may be simply done by using (180° — 1) instead of ( — 1). 



392 SPHERICAL ASTRONOMY. 

To find the time when the apparent conjunction takes place, let t denote the 
interval, in units of an hour, to be applied to the time of the true conjunction, and 
h the common hour angle of the bodies at the true conjunction. Then the 
position of the sun, not being supposed to be influenced by parallax, the common 
apparent hour angle of the bodies, at the time of the apparent conjunction, will 
be ti = h -f- 15° . t ; and therefore at this time, 

(cos I \ ' 

P' -1 sin (h + 15° . t), 
cos Df ' 

so that the conditi n for apparent conjunction, viz. a' = a — A a = 0, gives 

Ult ~ (^coTIj) siQ ( h + U °- t ) =° ( 21 ) 

for the determination of the interval t, which from this equation will be best found, 
perhaps, by the usual method of double position. We only want, however, an ap- 
proximate value, and may therefore avoid much unnecessary labor in estimating 
this time. Thus, at the time of true conjunction, the same approximate formulae 
may be adopted as used at page 3b5, viz. • 

™ COS I • 7 

A a = P sin h, 

cos 1) 

dh 



™ / dh . i( .\ cos/ 

A «i = P l — r- sin 1 ' I cos h, 

\dt / cos D 



dh 

m which — applies to the moon. It is evident, then, as the true positions of the 

bodies have no difference of right ascension, that A a is the apparent difference of 

right ascension ; and consequently, as the relative apparent motion in right as- 

™ /dh . „\ cos I , . 

cension is a! — a ai or a x — r I — — sin 1 ' I cos h, the correction t to be 

\dt / cos JJ 

applied to the time of true conjunction to get that of the apparent, will be 

_. cos I . , 

P t: sin A . 

cos D sin h 

t = 



-r,, /dh . ,\ cos I , cos I) /dh . ,„\ 

l i — P I ~r- sm 1" I cos h ai -=r. ; — I -z— sin V I 

1 \dt / cosi> * P' cos I \dt / 



cos h 



To facilitate the calculation of this expression, we ma}* use 57' as a mean value 
for P' and 14° as a mean value of D. Assume, therefore, 

100 cos I) _ 100 cos 14° _ [0.23103] 
1 ~ P' cos I ~~ 57 cos I cos / 

6 = 100 (— sin 1") cos h = [1.40274] cos h 

#"= 100 sin A 
for which the nearest whole numbers will suffice, and we shall have 



(22) 



-.,./-! (23) 

The values of the factor/ are given for various principal places in the table at 
page 40G : for any place not contained in that table it can be computed from the 
above expression, and used as a constant factor for all eclipses at that place. The 
Talues of 6, S( l », are also tabulated at page 405, where, for convenience, the argu- 
ment A is given in time. 



APPENDIX XI. 393 

II. — Formulae of Reduction to different Places 

Before quii ing this subject we shall give a method of calculating numerical 
equations which will serve to determine, with much ease and with sufficient accu- 
racy, the circumstances of an eclipse of the sun for any place comprised within a 
certain range of country. To effect this purpose in the most ample manner, in 
again proceeding with the general determination of the time of a phase, whose 
apparent distance of centres is a', we shall, in the expressions, separate as much 
as possible the quantities whi«ch involve the position of the place on the earth. 
The values of the co-ordinates x, y, given at p. 887, observing that a — a a = a', 
may be put down as follows : 

x = [(D + a' corr.) - S] — a D, 

y = a cos D' — A a COS D', 
and will thus consist of two terms, over the former of which the particular place 
on the earth has but little influence. If i denote, as before, the inclination of the 
apparent relative orbit, these ordinates resolved in the direction of n, perpendic 
ular to the orbit, and in the direction of the orbit, will give x cos i — y sin i, and 
x sin t -f- y cos i. It is evident, then, that x cos j — y sin t represents n, the near- 
est approach, and x sin t -f- y cos i the distance of the moon from it, which distance 
is estimated in the direction of her motion. At the time of the beginning or end- 
ing of the phase, the distance of the moon past the nearest approach, or greatest 
phase, will be ^ a' sin w; therefore the moon precedes this position by a dis- 
tance equal to ^f a' sin w — (x sin i -f- y cos t), which, divided by -^— , the hourly 

cos t J 

. . . . __ A' COS l . COS I 

motion in the orbit, gives ^£ sin w (x sin i 4- y cos t) for the inter- 

3/i yi 

val, in units of an hour, to be applied to the assumed time T to get the time t 
when the phase takes place. Assume, therefore, 

k = [3.55630]—— . (1) 

and, the time being counted in seconds, 

k 
t = l'-pks'mu) (x sin i -\- y gos t) (2) 

Also, x cos t — y sin i expressing the nearest approach, we evidently have 

x cos t — y sin t 



Make now the following assumptions: 

(D -J- a' corr.) — $ a cos D' 
p = ; cos t — sin l 

r A' • -A' 

k , k 
q = — ; [(D + a' corr.) — SI sin i -\ -a cos D' cos 

A A 

A D A « COS D' . 
A p = — cos i ; sin i 



k k 
A q = — : A D sin i -1 A a cos D' cos i 

A A 



(3) 



(4) 



(«) 



and k observing the above values of x and y, the equations (2), (3) will become 

coso>=p— Ap, ) ,. 

t = T '+ k sin u> — (q — A q) ) ' K 



394 



STHERICAL ASTRONOMY. 



Let y, \p be determined by the equations 

(-P + a' corr ) — $ " 

A~ 
a cos D' 



y cos </- 
Y sin xp = 



(?) 



(8) 



and p, q will take the following values: 

p = y COS (ip + l) ) 

5 = & j, sin (</, + i) J 

It yet remains to determine the values of A p, A 9, which depend on the po- 
sition of the place of observation. Adopting the notation used in the equations 
(3), (4), (9), (10), pages 379 and 382, we shall have 

[5.31439] A cos I . , 

A a = - — — . - sin A, 

1 — n cos JJ 

[5.31439] A \ . . cos (A + £ A a) 

A JJ = *=— : — I sin / cos D — cos Z sin i) 



J?== Lfi-31439J ^r 
1 — «i L 

To simplify the expressions, let 



cos •£ A 



^]- 



[5.31439] J. cos 2)' 
(I — n) a' * cos 2) ' 

[5.31489] -4 _ [5.31439]^ . _ 
c = f— . cos i>, a = — ^ — ■ . sin D ; 

(l-»i)A' (1-M,)A' 



and 



6 A ' cos I sin A 
Aa = Z^V ' 

A D = c A ' sin / — a A ' cos I 



cos (A -f I A «) 

COS i A a 



A a 



= c A ' sin Z — a A ' cos £ cos A + a A' tan cos Z sin A. 



A_a 
2 



6 sin 



sin A 1 



These substituted in (5) give 

A p = c cos t sin I — cos l\ a cos 1 cos A — ( a cos 1 tan — 

A q = k c sin t sin I — cos / 1 k a sin 1 cos A — (k a sin 1 tan \-kb cos i) sin A I 

COS -0' 

The value of b contains the factor — , for which we have 

cos D 

cos D' 

= cos A D (1 + tan D tan A D). 

cos D K ' 

Substitute the first value of tan A D, p. 379, and 



cos D' 
cos D 



= cos a D 



1 — p sin P — cos (h) 

cos JJ 



1 — p sin J 3 [sin Z sin J) -f- cos Z cos D cos (A)]' 



Or, putting A instead of (A) in the numerator, which cannot sensibly affect the 

value of the fraction, 

cos D' _ 1 — n 

^r = cos A JJ . . 

cos D 1 — wi 



APPENDIX XI. 395 

This, snpDosing cos A D = 1, reduces the values of the constants a, b, c, to the 
following : 

[5.31439] A ^ 

""(1-wO A' I (9) 

c = b cos D ; a = b sin D J 

A <* 
If e be a small arc determined by g cos e = 6, # sin e = a tan — — -, we shall have 

a cos t tan 6 sin i = g sin ( — i -f- <?) = g cos (90° + * — c) ; 

& a sin i tan 1- k b cos » = kg cos (t — e) = kg sin (90° + i — e). 

a 

However, as e must always be a very small arc, we may suppose cos e = 1 
also g = b, and, e being expressed in minutes, 

J_ a A_a_ a _ -, . ni)t , . 

60 6 2 120 6 L J v ' 

If therefore 

x = (90° + — • («) 

the values of A p, A <?, will be 

A p = c cos t sin I — cos I (a cos j cos h — b cos x sin h) ) 

A q = & c sin « sin £ — cos £ (& a sin i cos ^ — k b sin x sin h) J 
Assume now 

A = the longitude of the place, 4" east > — west. 
H=. the true hour angle of the moon, for the meridian of Greenwich. 

L' ' = c cos i \ 

y' cos ($' — H) = a cos i V (13) 

y' sin (i// — jET) = b cos x < 
i" = k c sin i \ 

y" cos («//'' — H) = Tasini [• (14) 

y" sin (<£" — ^T) = k b sin x ' 
and we shall have 

A p — L' sin I — y' cos £ cos (<// + A — IT) = L' sin J — y' cos / cos (^' + X), 
A y = L" sin £ — y" cos I cos (<£" -J- h — H) = i" sin / — y" cos I cos (tp" + A) > 
so that the equations (6) will become 

cos w = p — 2/ sin / -}- y' cos £ cos (»// -\-\) in n\ 

t = (T - q) T & sin a -f Z" sin / - y" cos Z cos (»/," -f A) ) ' " * ' 

After computing the constants k, p, q, L', L", *//, \p n , by means of the equations 
(1), (7), (8), (9), (10), (11), (13), and (14), we shall thus have two numerical equa- 
tions for the determination of w and the Greenwich time t of the phase, for any 
place whose latitude is I and longitude A. The accuracy of the determination will 
principally depend on the proximity of the resulting time t to the assumed time 
T; and therefore the result will be near the truth for all places where the phase 
will take place near to this time. 

In making these calculations for any particular portion of country, which for the 
partial phase will be necessary for both the beginning and ending, it will be best 
in the first instance to fix upon a place near the centre and compute the eclipse 
for that place, which computation will furnish good mean values for the data i>, 
«, 6, a corr, A D, A a, «, y h a', A, and comp. log (1 — n^). 



396 

By supposing 
the expressions 



SPHERICAL ASTRONOMY 



£' COS V = y, 
I' sin I' = - £', 



t cos I" = y" ) 



— U sin l-\-y' cos £ cos ((// -f-A), 

— L" sin Z + y" cos £ cos (xp" -f- A), 
will take the forms 

£' [sin r sin I -+- cos £' cos I cos (t£' -{- A)], 
£" [sin £" sin / -J- cos I" cos I cos (<//' -f- A)] ; 

and, without the factors £', f", will represent the cosines of the distances of the 
proposed place from two other places whose latitudes are /', /", and west longi- 
tudes xp\ \p". The former of these two places will be near to the southern pole of 
the true relative orbit, and the latter will be near to the orbit itself, and will pre- 
cede the moon by a distance nearly equal to 90°. 

For purposes which do not require great minuteness, the preceding equations 
will admit of some simplification by neglecting the small angle e. Add the 
squares of the equations (13) and (14), observing that c 2 + # 2 = & 2 , and 

L" 2 + y' 2 = b 2 (cos 2 1 + cos 2 x ), 
L" 2 + y" 2 = k 2 b 2 (sin 2 1 + sin 2 x ) ; 
which give the general relation 



L" 2 y' 



= 2b 2 



k 2 ' k 2 

By neglecting e, x = 90° 4" l , cos % = — sin i, sin x = cos i ; and then 

L' 2 + y' 2 = b 2 , 
Z" 2 +y" 2 = k 2 b 2 ; 

which united with the equations (16) give I = b, £" = Jc b, and hence 



sm/ = — — = 



= — cos D cos i ; 



sin (xp' 



Y ~ 
H) = 

sin I" = 



$' cos V ■= b cos V ; 
b cos x 6 sin i 



Y 

f 



b cos £' 
& c sin i 



Jcb 



sin t 

cos /' ' 

= — cos D sin i 



sin (<//' — H) = 



y" == f COS I" = kb COS J" 

& 6 sin x _ & 6 cos * 

y" ~~ k b cos £" 



cos I" 



Or, 



Z' 



Z" = 



sin V = — cos D cos t 
- 6 sin I' ; y' — b cos /' 



sin (*// — H) = 



sin » 



sin J" = ■ 
A 6 sin Z" 



cos /' 
cos D sin t 

y" = kb coa J" 



sin 



W-B) = 



cos 



*" 



(1*) 



(18) 



(19) 



APPENDIX XI. 



397 



These may be employed instead of the equations (13) and (14) ; or the equations 
(13) and (14) may be adopted in their reduced form, viz. : 



= cos D cos 



cos (i// — H) = sin D cos t 



—~ sin (*// — H) = — sin t 
b 

— - = cos D sin i 
k b 



JLcob^"-^) 



D sin 



/- sin U" — H) = cos i 
a; o 

in which the coefficients c, a, will not be required. 



(20) 



(21) 



III. — Transits of Mercury and Venus over the Disk of the Sun. 



These phenomena are, in many respects, analogous to that of an annular eclipse 
of the suu, and admit of a similar calculation ; the principal distinction consists in 
the negative sign of the relative motion of the planet in right ascension, which 
will make the inclination of the orbit always obtuse, and therefore render some 
modifications necessary in the determination of the particular species of the other 
angles which enter into the computation. To avoid any confusion that might 
thus arise, we shall adopt the sun as the movable body, and refer his positions to 
that of the planet which we now suppose to be stationary. Thus, 

6 = the 0's declination; 
D = the planet's declination ; 

n = the 0's equatorial horizontal parallax; 
P — the planet's equatorial horizontal parallax ; 

a = © 's right ascension minus that of the planet ; 

x — (&' 4" a ' corr.) — D ; 

y = a' cos <5' ; 

x Y s= the O's motion in declination minus that of the planet ; 

2/i= (0's motion in right ascension minus that of planet) . cos 6' ; 

and so we might proceed as with an eclipse of the sun, only observing that the 
relative parallax p (ir — P) is a negative quantity, and that the positions of the 
contacts on the limb of the sun, as in the case of an occultation, will be at points 
opposite to those which come out in the calculation. However, as the relative 
parallax is always very small, the ingress and egress of the planet will be seen at 
all places on the earth at nearly the same absolute time ; it will, for this reason, 
be best to compute first the circumstances for the centre of the earth, and then to 
ascertain the small variations produced by parallax for any assumed place on the 
surface, which may be readily deduced from the preceding equations for the reduc- 
tion of an eclipse of the sun. Let w, (t), be the values of w, t, for the centre of the 
earth, and, by separating the effects of parallax from the equations (6), 



108 



SPHERICAL ASTRONOMY. 



cos w = p, 

(t) = (T— q) =F k sin w, 

A cos w = A jo, A t = — A j T ^ A sin v. 

But, as the quantities A cos w, A sin w are very small, A sin w =— A cos w — 

u J sin w 

cos w 

that is, A sin w = — A p . Therefore, 

sin w 



COS W / . COS W \ 

t = — A q ± k Ap - = ± 1 A A p- TAyl 

sin w \ sin w / 



A t = 



In this expression substitute the values of A p, A q, according to the equations 
(12), and we find A t = 

f cos[— iTw] . / cos[— iTw] cos[-x=Fw] . 

±1 &c -. sml—coslika -. J cosh—kb -. 

sin w \ sin w sin w 



cos &— k b """ L . ^ ' " J s i n hjh 



in which 6 



p(rr-P) 



p(P-*) 



6 ccs 3 and a = 6 sin & 



Because of the smallness of the parallax, the angle ewill not be appreciable, an-1 
consequently x == 90° + l , cos [ — X T w] = sin [ — i T w]. "We shall therefore 
have for the time of ingress or egress the following general expression, in which 
the terms within the brackets depend on the position of the place of observation ; 
also the upper signs apply to the ingress, and the uwer signs to the egress. 

I = T — q =F k sin w 



=F k b I cos <5 

Assuming k" 



cos [— «T w] 



sin w 

— kb 



(. ,cos[— (TWJ 
sin o -. - cos h— 
sin w 



8in[— iT'W'l 



t)cQS*l 



p sm w 



, this expression will resolve into the following : 



tan t = — 



k = [3.55630] 



A cos t 



y cos rp = 

y sin xp = 



(6 + a corr.) — D 

A 
a cos 6 



COS W = y COS (li/ + t) 

g- = & y sin (i// -f- «) 
(*) = T— q T A; sin w 

A sm w 

— as cos [ (— t) T w] cos S 

y" 

■jr, cos ty" -fi) = cos[(-i)Tv] sin * 

^ ,in ( " — H s sin [ (- i) =F w] 



(a) 
(6) 



(4) 



= (0T[y"pco 3 oos (^/' +*)-£"/> sin ] ft 



APPENDIX XL 399 

In these equations, 

H= the O's true hour angle from the meridian of Greenwich, at the time (t). 

For i exteri0r I contact of limbs, A = i * + * \ 
( interior ) ( a — s ) 

For contact of centre of planet with 0's limb, A = o-; 

s denoting the true semi-diameter of the planet, and a that of the sun. 

The equations (a), (b), (c), (d) will serve to determine the constants (t), y", Z" t 
xp", for the times of ingress and egress, and then there will result two numerical 
equations of the form (<?) to reduce the phenomena to any place on the earth's 
surface. 

For the points on the limb of the sun, we shall have 

At \ ingress I , angle from N. towards E. == \ ( 1S0 ° ~ '} ~ W I for direct image. 
( egress ) ( (180 — t) + w ) 

(-«) T .W 



, for inverted image. 

which will be sufficiently accurate for all places on the earth. 

The time ^may be assumed near to the time of conjunction in longitude, or 
right ascension, as it may suit convenience. For Mercury, if very minute accuracy 
is wanted, it may be necessary, for more correct values of (t), to assume two times 
SPnear to the times of ingress and egress; but it is very questionable whether 
such a precarious extent of accuracy would sufficiently recompense the time ex- 
pended on the calculation. 

IY. OCCULTATIONS OF STARS BY THE MOON. 

These may be calculated in the same manner as eclipses of the sun, the only 
difference in the operation consisting in the star having neither motion, parallax, 
nor semi-diameter. But where great minuteness is not wanted, these particular 
circumstances will afford some degree of simplification to the expressions, if that 
parallax of the moon be adopted which would answer to the star as an apparent 
place, since this parallax, at the times of immersion and emersion, will then be 
precisely that of the respective points of the moon's limb which come in contact 
with the star; and thus the augmentation of the moon's semi-diameter will be 
evaded, so that the true semi-diameter may be employed. For this novel and ju« 
dicious expedient we are indebted to Carlini. — See Zach's Correspondence, vol. 
xviii., page 528. 

As in the case of the sun, let <5 denote the declination, and h the hour angle of 
the star, and let P represent the equatorial horizontal parallax of the moon. 
Then, for the effects of parallax in right ascension and declination, we must sub- 
stitute S for D\ and h for h in the formulae (2) at p. 379, which thus become, dis- 
regarding \ A h, 

x> cos I . , 

A«=/ir =: sin h, 

cos B 

A Z> = p P (sin I cos $ — cos I sin 5 cos h). 

As soon as the immersion takes place, these expressions will represent the paru 

of that point of the moon's limb which is in contact with the star; and thereftn 

the application of this parallax to the centre of the moon will produce an apparent 

distance A ' of the centres, equal to the true semi-diameter s of the moon. Also 

as the star, in the course of the occultation, is only affected with its apparent diar* 

nal motion, the hourly variations of the above values will be 



400 



SPHERICAL ASTRONOMY. 



r>/ dh ■ , 
A a x = p r I — sin 1 

<dh 



) 



cos I 



cos A. 



dh 



A Di=pP I — sin 1" 1 cos Z sin <5 sin A ; 



in which — is 15° 2' 28", the hourly diurnal motion of the earth, and therefor*) 



dh . „ r 

— sin 1" = T9.41916]. 

d t 



Assume 



l ) = D cos 1 = 



V 



1 — e sir 



&( 2 ) = p sin ? — 



v/r 



sin 2 I' 



dh 



(1) 



^ 3 )= p eos I — sin 1" = [9.41916] 

which are constant coefficients depending on the latitude of the place ; then 

<I>W . P 



«^>.P . . 
A a = I ^- sin A, 



A aj 



cos A. 



(2) 



cos D cos D 

A D = (0( 2 ) cos (5 — 00) sin 8 cos A) . P, A A = tf> (3) . P sin 3 sin A. 

If, in the values of A a, A oi, we use cos 8 instead of cos P, the values of x, y, z u 
$i, p. 387, will become 

x = (Z> — 8) — (<pW . p cos 6 — 0(i> . P sin <5 cos A) " 

t/ = a cos 3 — 0( ! ) . P sin A 

Xl ~Di — <p& . P sin <5 sin A 

yj = a 2 COS <5 0< 3) . P COS A 

in which we have disregarded the a correction. 

With the values of x, y, xi, yi, so found, we may then proceed with the equa- 
tions (16) and (18), pages 387 and 388, as in the case of a solar eclipse. 

This method is similar, and, as far as accuracy goes, the same as the recent 
method of Professor Bessel, who divides all the quantities by the equatorial hori- 
zontal parallax of the moon. He assumes 
a cos S 



P ' 
D — 8 



ai cos 8 



P, 



q=z 



q ~P 



(3) 



in A ) 



u = cpW sin h, v! = <p( 3 l cos A 

v = 0< 2 ) cos 8 — <pW sin 8 cos A, v = <p' z ) sin 8 sin 
so that if we change the signification of the symbols x, y, x lf yi, and suppose them 
now to represent the preceding values divided by P, we shall have 

x = g — v, Xi^zg' — v')^ ^ 

y —p — u, t/i =zp — u' ( ' 

These values being adopted, in proceeding with the equations (16) and (18) we 



(4) 



must use A' — •— , the value of which, according to Burckhardt's Tables de la Lune 

(Paris, 1812), p. 73, is [9.43687]. Much facility is thus given to the calculation 
pf occupations, for different places, if the values of p, q, p', q', which are indepen- 



APPENDIX XI, 



401 



dent of geographical position, are published ; but if these quantities are to be pre- 
pared by the computer, the equations (2) will be more simple and 

The chief difficulty in the calculation of occupa- 
tions, for any particular place, rests in the selection 
of the list of stars : in the course of any year a great 
number will be liable to occultation on the earth 
generally, though the majority of them will not be 
occulted at the particular place for which the special 
calculations are to be made. It will therefore be 
expedient to reject such stars as may at different 
stages of the calculation be shown to violate any 
conditions necessary for the existence of the occulta- 
tion, its appearance above the horizon, or its exemp- 
tion from the glare of sun-light For the general 
list we may observe, that the difference of declina- 
tion at the time of conjunction must be within the 
limit of about 1° 30', and that all stars, whose con- 
junctions with the moon occur within two days of 
new moon, may be omitted. In the process of exclu- 
sion for the particular place, the first and most pal- 
pable condition is, that at the time of conjunction 
the sun must be below, or near to, the horizon ; if 
more than half an hour above the horizon, the occul- 
tation will surely be useless; another condition is, 
that the star must be above the horizon ; and, to 
satisfy this, the hour angles at the times of immer- 
sion and emersion must be less than its semi-diurnal 
arc. The value of the hour angle at the time of 
apparent conjunction may be determined by increas- 
ing that at the time of true conjunction by the quan- 

tity , according to the tables on pages 401 

ai •/ — 6 
and 402 ; and it may be observed that this hour 
angle must not exceed the semi-diurnal arc by more 
than half an hour. For the latitude of Greenwich, 
the semi-diurnal arcs, allowing 33' for refraction in 
the horizon, are shown in the annexed table. 

As a final test for the exclusion of unnecessary 
stars, it is useful to calculate the extreme limits of 
latitude between which the star will be visibly oc- 
culted on the earth. These will evidently appertain 
to the extreme northern and southern points of the 
northern and southern limits of contact, determined as for a solar eclipse, a point 
in the northern or southern limit will depend on the formulae Nos. 27, 28, pages 

359-60. Thus, 

n ±A' 
cos w = — =- , 



Dec. 


Semi-diurnal Arcs, for 


the Lati 


Aide of 


of 


Greenwich. 


Star. 










Dec. North. 


Dec. South. 


o 


h. m. 




h. m. 


o 


6 4 




6 4 


i 

2 


6 9+ 
6 14 


5 
5 
5 
5 
5 


559 5 
5 54 D 

5 4 9 I 

5 43 6 

5 38 5 


3 
4 


6 19 
6 24 


5 


6 29 


6 


6 34 


5 
5 
5 
6 


5 33 5 


7 


6 3 9 


5 28 I 


8 


6 44 


5 23 5 


9 


6 5o 


5 18 5 


IO 


6 55 


5 
5 


5i3 5 


ii 


7 ° 


5 7 6 


12 


7 6 


6 


5 2 * 


i3 
i4 
i5 


7 Ir 

7 17 
7 23 


5 
6 
6 
5 
6 
6 


4 56 <> 

4 5i : 
4 45 I 
4 4o I 

42 8 ; 

4 22 

4 i5 I 
4 9 6 


16 


7 28 


17 


7 34 


18 


7 4o 


l 9 

20 


7 47 
7 53 


7 
6 


21 


8 


7 


22 


8 6 


6 


4 » I 

3 56 6 


23 


8 i3 


7 
8 


24 


8 21 


li 9 « 

3 4i 


25 


8 28 


7 


26 


8 36 


8 


3 34 I 
3 26 8 
3 18 8 


27 


8 44 


8 


28 


8 53 


9 


29 


9 2 . 


9 


3 9 9 


3o 


+ 
9 12 ' 


IO 


3 o-9 



sin Z — 



cos w 



M=— t ± 



and thence, 



sin D' cos Z + cos D' sin Z cos M. 



26 



402 SPHERICAL ASTRONOMY. 

It is now our object to ascertain what value of </ will render the value of /, so 
deduced, a maximum or a minimum, and what will be the corresponding value of /. 
Let <f> be an arc determined bj the equation, 

cos Z=cos sin w (6) 

Then by uniting with it the equation 

cos w' sin Z = cos w . . . . . . (7) 

we infer that 

sin (d' sin Z = sin <p sin w . (8) 

because the squares of these three equations added together will give unity on each 
side. By these equations we shall hence have 
sin D' cos Z = sin D' cos <p sin w, 
sin Z cos M = sin Z (cos t cos w' T sin » sin »'), 

= (cos w' sin Z) cos { T (sin J sin Z) sin «, 
= cos i cos w =F sin t sin sin w ; 
and, consequently, 

sin I = cos D' cos t cos w -j- sin w (sin i)' cos <p 3= cos i)' sin i sin ^), 
which now involves only one variable <p. Again, assume two arcs, 6, tp, which will 
fulfil the equations, 

cos cos xp = sin D' ........... (9) 

cos sin xp = ± cos 2)' ein i (10) 

A third equation will follow from these, viz. : 

sin = cos D' cos « (11) 

because, as before, the squares of these three equations will together make unity. 
The value of sin I will now become 

sin I = cos w sin -f- sin w cos cos {<p -f- xp). 
The angle <p -|- t/> being the only variable in this expression, it is evident that the 
greatest value of / will have <p -(- \p == 0, and the least -f- ^ = 180°. Therefore, 

!e r :f st } ^"° ° f ' = i : t z i • -* - f - 1 zs } ** 

These would be the extreme latitudes for the appearance of the occultation if tie 
earth were a transparent body ; as this, however, is not the case, it will be neces- 
sary that the star should be above the horizon, a condition not included in the 
preceding equations. The zenith distance Z must not exceed 90°, and therefore 
cos Z must necessarily be a positive quantity. 

By the equation (6) cos Z must have the same sign as cos <p, and this must be 
the same as -J- cos xp for northern limit, or — cos \p for southern limit, because in 
the former case <p -{• ip = 0, and in the latter <p + ^ = 180°. But, by (9), cos </- 
must have the same sign as D'. Consequently, 

For < ,i > limit, cos Z has the same sign as J ^ ™' 

It is evident, therefore, that the extreme northern limit will have the star below 
the horizon, and be excluded when D' is negative, and that for the same reason 
the southern limit will be excluded when D' is positive. Thus the only admissi- 
ble extreme limit will be determined by the equations 

cos w = H , lib=d±vr . ... (12) 

• ■* 

upper signs when D' is positive, and under signs when D' is negative. 

The other limit for the actual appearance of the occultation wil evidently be ono 



APPENDIX XI. 



403 



of the two places where the other limiting line meets the rising and setting limits, 
and will be determined by 



n T A 



sin U = eos D' cos [ (— t) T w] 



(13) 



using, as before, upper signs when B' is positive, and under signs when D' is neg- 
ative. 

The equations (li), (12), (18), for convenience in determining the species of the 
angles, may be put in the following form : 

T »+ A' 1 



COS Wi 



T n 



COS w 2 



p, . — ■■- F , 

sin 9 = cos D' eos t 

h = Wi — 
sin h = =F cos 2)' cos (w 2 — i) 

observing that Wi, w 2 , 0, and t, must here take the same sign as D' ; also, 



(14) 



upper ) • , 7v . S positive, 

y '. > signs when i? is 1 ' t . ^ 

under ) => ^ negative 



These formulae are applicable to a solar eclipse. For an occultation of a 
star by the moon, P' will be the moon's horizontal parallax, and a' her semi- 
diameter, which, as these limits are not wanted very accurately, may be regard- 
ed as true quantities ; also, we may neglect n and so take <5 instead of J)', Sinco 

— = [9.43587] = .2725, the formulae for an occultation will hence be 



tan 



_J)j__ 

ai Cos 6 



cosW!= T — — .2725, 



n = (cliff, dec.) cos t 

~ +.2725 



J, = Wi — < 



(15) 



COS W 3 = 

sin 9 = cos S cos < 

sin i 2 =■ J- cos 6 cos (w 2 — t) 

in which we also give to the angles Wi, w^, «, 0, the same sign as 6, and use upper 
signs when 6 is positive, and under signs when 6 is negative. We may also ok- 
serve, that, 

1. When <5 is north, l t is the most northern limit ; and when 6 is south, li is the 
most southern limit. 

2. When Wi is imaginary, h will be 90°, ami of the same name as <?. In this 
case the occultation will be visible about the pole of the earth, which is presented 
to the star ; the visibility will extend beyond the extremity of the disk of the earth 
as it would be seen from the star. 

3. When w 2 is imaginary, li will be the complement of &, and of a different name 
from 6. In this case, if we consider the disk of the earth as seen from the < tar, the 
visibility of the occultation will extend beyond that extremity of the disk which 
has the pole on the other side of it. 

After an occultation is computed for any particular plac»\ if we deduct the star's 
right ascension from the sidereal times of immersion and emersion Ave shall get 
the hour angles of the star, -f- West, — East. By comparing these hour angles 
with the semi-diurnal arc of the star, we can distinctly ascertain the positions of 
the star with respect to the horizon. 



404 SPHERICAL ASTRONOMY 



V. — Eclipses of the Moon by the Earth's Shadow. 

These may be also resolved in the same way as those of the sun. The absolute 
positions of the moon and shadow being independent of the position of the specta- 
tor on the earth, the determination of parallaxes will be here unnecessary, which 
much simplifies the calculation of these eclipses. The considerations requisite to 
be attended to, by way of distinction, are the following : 

Semi-diameter of the shadow = — (P' -f- * — a ). 

Semi-diameter of the penumbra = — (P' + » — a) + 2 ff . 

Right ascension of centre of shadow = that of the sun ± 12 h . 
Declination of centre of shadow = that of the sun with a contrary name. 

The figure of the earth being spheroidal, that of the shadow will deviate a little 
from a circle, so that, to have a mean radius, the horizontal parallax of the moon 
must be reduced to a mean latitude of 45°. This will give 

P' = [9.99929] P: 
P denoting the moon's equatorial horizontal parallax. 

Also, 

a = right ascens. moon minus right ascens. centre of shadow ; 
x ■=. (dec. moon -f- a corr.) minus dec. centre of shadow ; 

y ■=. a cos D. 

With these we compute according to the equations (16) and (18), pages 387 and 
388, observing the following values of A' : 

For -J f , ern ^ j- contact with shadow, A' = semi-diam. shadow ± s. 

For •< ? x , e , [■ contact with penumbra, A' = semi-diam. penumbra ± *. 

The angular positions of the points where the contacts take place will be esti- 
mated on the circumference of the shadow or penumbra the same as they were 
before on the limb of the sun. These angles will therefore be in a reversed posi- 
tion on the disk of the moon, and consequently as they come out from the compu* 
tation will have reference in the first instance to the inverted appearance of the 
phase. 

The relative orbit of the moon, not being affected with parallax, will not sensibly 
deviate from a great circle in the course of the eclipse ; and hence the assumption 
of the particular time, on which to found the calculation, will be but of little im- 
portance: any convenient time may be assumed near the time of opposition. 

It will be unnecessary to add any further remarks. "We shall conclude this pa- 
per with a tabular recapitulation of the formulae which relate to the phenomena 
for a particular place, in which eclipses of the moon, for the sake of clearness, are 
given separately. The object of this taMe, like the former one for the general 
eclipse, is to simplify and expedite, by an jasy reference, the actual operations of 
the computer. 



APPENDIX XI. 



405 



I. ECLIPSE OP THE SUN FOR A PARTICULAR PLACE. 

1. h = apparent time of true (5 in R. A. to nearest minute. 

With this as an argument, take out the numbers 6, S (l \ from the following 
table : 



Table for reducing the true to the app. <$ in 


R. A. 






6 


fid) 




6 


00) 


Hour Angle 






Hour Angle 














h 




same 


h 




same 


at true <5. 


+ — 


sign 
as h. 


at true <j>. 


+ — 


sign 
as h. 


h. m. 


h. m. 






h. m. 


h. m. 






o o 


12 o 


25 


o 


3 


9 ° 


18 


7i 


10 


ii 5o 


25 


4 


10 


8 5o 


17 ■ 


74 


20 


4o 


25 


9 


20 


4o 


16 


77 


3o 


3o 


25 


i3 


3o 


3o 


i5 


79 


4o 


20 


25 


n 


4o 


20 


14 


82 


5o 


IO 


25 


22 


5o 


10 


14 


84 


I 


II o 


24 


26 


4 


8 


i3 


87 


IO 


io 5o 


24 


3o 


10 


7 5o 


12 


89 


20 


4o 


24 


M 


20 


4o 


11 


9 1 


3o 


3o 


23 


38 


3o 


3o 


10 


92 


4o 


20 


2.3 


42 


4o 


20 


9 


9 4 


5o 


IO 


22 


46 


5o 


10 


8 


9 5 


2 O 


IO o 


22 


5o 


5 


7 


7 


97 


IO 


9 5o 


21 


54 


10 


6 5o 


5 


98 


20 


4o 


21 


57 


20 


4o 


4 


98 


3o 


3o 


20 


61 


3o 


3o 


3 


99 


4o 


20 


r 9 


64 


4o 


20 


2 


100 


5o 


IO 


l 9 


68 


5o 


10 


1 


100 


3 o 


9 ° 


18 


71 


6 


6 





100 



Then. T denoting the approximate mean time of app. <3, in units of an hour, 

6(1) 



T= mean time true <3 4- 



■/- 



in which en must be used in minutes of arc : also / = -— — -^, is a factor depend- 

cos I 

ing on the latitude, which, for several principal observatories, is, for convenience 

included in the following table : 



406 



SPHERICAL ASTRONOMY. 



Auxiliary Quantities depending on Geographical Position. 


Place. 


P 


cot I 


cos I 


/ 


Longitude. 


Aberdeen 


9.99900 


+ 9.81289 


9-7363 7 


3-12 


h. m. s. 
W. 8 23 


Altona . . 


9 


99908 


+ 9-87133 


9.77576 


2-85 


E. 39 47 


Berlin .... 


9 


999.0 


+ 9-88 7 5i 


9.78603 


2.79 


E. 53 36 


Bedford .... 


9 


99911 


+ 9-89345 


9-78974 


2.76 


W. 1 52 


Cambridge . 


9 


999.1 


+ 9-89231 


9-78903 


2.77 


E. 24 


Cape of Good Hope 


9 


999 56 


— 0-17494 


9-91980 


2-o5 


E. r i3 55 


Dublin .... 


9 


99909 


+ 9 -8 7 3b5 


9 -77737 


2-84 


W. 25 22 


Edinburgh . 


9 


99902 


+ 9-83256 


9>75ooi 


3-o3 


W. 12 44 


Greenwich . 


9 


999.3 


+ 9-90381 


9.79610 


2.72 


000 


Orm&kii'k 


9 


99908 


+ 9.87092 


9-77549 


2-85 


W. 11 36 


Oxford .... 


9 


99912 


+ 9.89939 


9-79340 


2.74 


W. 5 2 


Kensington . 


9 


999.3 


+ 9 • 90340 


9.79586 


2-72 


W. 47 


Milan .... 


9 


99928 


+ 9.99577 


9-84736 


2-42 


E. 36 47 


Paris .... 


9 


99920 


+ 9.94451 


9-81997 


2-58 


E. 9 22 


Slough .... 


,9.99913 


+ 9.90337 


9-79584 


2.72 


W. 2 24 



2. The time T being computed to the nearest minute, take out the correspond- 
ing values of P, x, c } 3, from the Ephemeris ; and prepare the constants 

e = [4.68555] p, 
A = c (P — *), m = A cos /, 

Q i = [4.7172], Q2 = mQi am i t 

s = [9.43537] P. 



3. Take, out D, 3, a, D h a u for the time T. 

h = sidereal time at place minus J> 's right ascension, to the tenth of a minnt^ 
In arc. 

m 



k = 



n = k cos h, 



cos D' 

A a = [5.31439] k sin A [corr. for »], 

A ai = Qi n, AA= Q2 sin A. 

Correction for n to be taken from the table on page 383. 



tan = cos (A) cot /, G = cos (A) cos I, 

sin 



tan M x-n 



cos (0 -f- D) 



tan (A), 



tan « = tan (0 -\- D) cos M t 



B = cos M cos t ; 

sin G 



CheGk - * '^{6 + J))-B- 
M to be in the same semicircle with h. 



APPENDIX XI. £07 

n, = A sin e, aD = [5 31439] A B [corr. for «i], 

s' = s [corr. for wi]. 

( total or annular ) ( s' ~ o ) ' 

Correction for % to be taken from the table on page 383. 



D'=i> — A A 


a' = a — A a, 


«/ = (a — A a) COS D', 


^ = (aj — A <*i) COS 2)', 


# =. (J7 -j- a' corr.) — $, 


*! = A — A A. 


tan £ = -, 

X 


cot l = ^, 


sin # 


a; 
cos S' 


» = IFcos[ — (5+t)], 


W cos t [3.55630] 

yi 


cos U = — -, 

A' 


cos w 



ft 

«'=[—08f + «)]—. * = [-(0 + ')] + ■». 

^ = c sin a, fa = c sin 6. 

Timeofj be ^ nnin gl=7'+^ 1 l. 
< ending J ( fa ) 

Time of greatest phase = £ sum of times of beginning and ending. 

When n < s' ~ <r, the eclipse will be total if s' > o, or annular if s' < a; in this 
ease these last equations No. 7 must be repeated for this phase with A' = s' ~ <r, 
the results of which ought to give the same time for the greatest phase. 

Take A' for partial phase, and 

Portion of sun's disk eclipsed = A ' — n. 

Magnitude of eclipse = , the sun's diameter being unity. 

8. For the positions of the points of contact on the limb of the sun, 
At | egwninf ( . ang j e from north towards east = j ' ~ l > ~ w j. for direct image. 

At \ inning j. a , e from north towards east = j (180- -*■ .)-. * ) for inverted 
( ending )' & ((180° — 0+«) image. 

For the position of the moon's centre at greatest phase, 

Angle from \ >■ towards east = \ ) 7 [ for direct image. 

( vertex ) ( ( — t) — M ) 

Angle from \ y towards east = \ ) „ V ,^ Mor inverted image, 

I vertex) ((180°— t)— If J * 



408 SPHERICAL ASTRONOMY. 

9. For a more accurate calculation of the time, &c, of beginning of the partial 
phase, assume a convenient time near to the preceding determination. For this 
time, take out the quantities D, D u S, a, a u from the Ephemeris; and proceed as 
in Nos. 3, 4, 5, 6, 7, omitting b, t^ and the times of greatest phase and ending. 

Let M\, «i, »i, be the values of the angles in this computation ; then, for the po- 
sition of the point of contact on the limb of the sun, 

Angle from \ (• towards the east = \ \ *V Wl -■£ for direct image. 

6 I vertex ) ( ( — iA— w i — Mi ) * 



Angle from \ n0rth \ towards the east = \ < 180 ° ~ '0 ~ Wl - I for inverted 
g ( vertex ) (180° — tA — Wl — JE/i J 



(180° — n) — ( 

(180° — ii) — wj — Mi ) image. 



10. For a more accurate calculation of the time, <fcc, of ending of the partial 
phase, assume a convenient time near to the first determination. For this time, 
take out the values of D, D l} <5, a, ai ; and proceed as in Nos. 3, 4, 5, 6, 7, omitting 
a, ti, and the times of beginning and greatest phase. 

Let M 2 , i 2 , » 2 , be the angles in this computation ; then, for the position of the 
point of contact on the limb of the sun, 

Angle from < C towards the east = ■< ) l2 { , 0)2 ,, [■ for rfire^ image. 

° (vertex) ( ( _ , 2 ) + W2 _ Jf 2 J s 

. , -. (north ) . , ,, . W180° — «s) + w * ) for inverted 

Angle from < J- towards the east = •< ),„ ■ ' __ /■ • „ M 

s * vertex ) ( (180° — * a ) + «*a — Jft ) image. 

II. FORMULAE FOE REDUCTION TO DIFFERENT PLACES. 

11. Instead of Nos. 5, 6, 7, substitute the following: 

D' = D — A D, a' = a — A a, 

Xi = Dt — A Di, y\ = (ai — A ai) cos D\ 

Xi r ., A ' cos * 

tan i = -, Jc = [3.55630] , 

2/i 2/i 

(D + a'corr.) — b . a cos D' 

y cos ip = J ; — - , y sin ip = ; — , 

jo = y cos (;// -{- ')> q = ky sin (<// + i), 

y— ? = T. 

[5.31439] A _ , _ 

12. 6 = r- — [corr. for n{\, 

A L J 

e in minutes = [7.9208] A « sin D, x = (90° + — *• 

18. if = the true Greenwich hour angle of D at the time T. 

-— = cos D cos i, TT = cos D ^ *f 

6 to 

y 7 v" 

' cos (t// — H) = sin D cos t, -~ cos (^" — H) = sin i> sin i, 

~ sin (i// — -H") = cos x, rr sin (y/' — //) = sin x . 

p « o 



APPENDIX XI. 409 

14. The constants T', k, p, L', L", y', y", being so computed, the angle w and 
the time t of the phase for any place whose north latitude is I and east longitude 
A, will be determined by the two following equations, in which the upper sign re 
lates to the beginning and the under sign to the ending. 

cos u) = p — U sin I -f- y cos I cos (A -j- ^/) ; 

t = T' =F k sin w -f- L' sin I — y" cos I cos (A -}- <j/'). 

The result will be the most accurate when the place is near to that on which 
the previous part of the calculation is founded. 

m. TRANSIT OF MERCURY OR VENUS OVER THE DISK OF THE SUN. 

(Same notation for the planet as for the moon.) 

15. Assume the time T near to the time of conjunction in longitude, or right as- 
cension. 

a = sun's right ascension — planet's right ascension in arc; 
o! = hourly variation of a ; 
Di = sun's hourly motion in declination minus that of the planet. 

For \ exter !° r I contact of limbs, A = \ ° + S ' 
( interior ) ( o- — s. 

For contact of planet's centre with sun's limb, A = <r ; 

tan , = A 

a x COS ft 

., A cos i P — ir 

h = [3.55630] 



ax COS <5 A 

(<5 + a coir.) — D . a cos $ 

y cos ilt = , y sin li = , 

r A A 

cos w = y cos (t// -f- 0, q — ky sin (i// -f- *)• 

16 ^T= the true Greenwich hour angle of at the time T. 

kb 



k" 



-— = cos <5 cos [( — i)| w]; 
/c 

y" 

-rjj cos (\p" — H) = sin 8 cos [( — i) qp w] ; 

K 

y — sin (<//' — H) — sin [( — T w]. 

17. Then, for the centre of the eai-th, 

(t) = (T— g) T k sin w; 
and, for any place whose latitude is I and east longitude A, 

t = (t) T [y" p cos £ cos (A -f. »//') — L" p sin ^ 
usinff" the upper signs for the ingress, and the under signs for the egress. 

The positions of the points of ingress and egress, estimated from the north point 
of the sun's limb towards the east, as the transit would be seen from the centre 
©f the earth, will be determined in the same manner as for the immersion and 



410 SPHERICAL ASTRONOMY. 

emersion of an oceultation, No. 19, using w for w. These angles may be assumed 
to be the same for any place on the surface, the effect of parallax being so very 
minute. 

IV. OCCTJLTATION OF A STAR BY THE MOON. 

General Limits of Latitude.. 

18. (a x and D t at true 6 )• 

tan i = ■ — -i— , n = (diff. dec.) cos i, 

ai cos S 

cos Wi = =f — — .2725, cos w 2 = ^ - + .2725, 

sin 6 = cos 8 cos j, 
h = Wi — 9, sin h = ^p cos 8 cos (W2 — *), 

•Wj, W2, i, 6, same sign as 8, 

U PP er I signs when 8 is i P osit ! ve > 
under ) & ( negative. 

"When Wi is impossible, £i = 90°, with the same name as 8. 

When W2 is impossible, h = complement of 8, with different name from i. 

Calculation for particular Place. 

(9. For the latitude of the place prepare the constants 

<j>0) = p cos I, f 2 > = p sin I = — <p'V = [9.41916] tf l \ 

■which will serve for all occultations at that place. 
Fo? the time of true <3 find 

h = sidereal time at place — right ascension of star ; 
and thence determine the time T, as in No. 1. For this time take out the quanti- 
ties P, s, D, D h a, a x ; and compute 

x =(& — $) — (0(2) . P cos 8 — f l > . P sin 8 cos h) ; 
y = a cos 8 — <p l ) P sin h ; 
x x — D x — 0' 3) . P sin 8 sin /< ; 
yi = a : cos 8 — tiW . P cos h. 
With these proceed as in Nos. 6 and 7, using A ' = s == [9.48537] P. 

20. For the positions of the points of immersion and emersion on the limb of the 
moon, 

At \ . [ , angle from north towards east= \ \ ~ l ;~ m r for direct image. 

( emersion ) 5 ((180°-i) + w ) 

At ■] . [■ , angle from north towards east= \ ; ' ~~ W r for inverted image. 

( emersion ) ( ( — t) + w ) 

For the same angles from the vertex we must deduct the parallactic angle for 
each time. 

21. If an accurate calculation is wanted, proceed as with a solar eclipse. 



APPENDIX XI. 411 



V. ECLIPSE OF THE MOON. 



11. Fix on a convenient time near to the time of opposition in longitude, or full 
', <oon; and for this time find P, s, n, a, 

a = ]) 's right ascension minus (©'s right ascension ± 12 h ), in arc ; 

a x = hourly motion of a ; 

x = ( D 's dec + a corr.) plus O's dec. ; 

Xi = hourly motion of x ; 

y = a cos J) 's dec. ; 

?/! = ai cos D 's dec. 

P' = [9.99929] P. 

Semid. shadow = — (P r -\-ir — a), 

fil 
Semid. penumbra = — - (P' -\-ir — a) -j- 2 <r. 
60 

For -J ? x , el na j [• contact with shadow, A' = semid. shadow -J _ f *• 

For \ ? x ern ^ (• contact with penumbra,. A ' = semid. penumbra ■] _ f s> 
The remaining computation as in Nos. 6 and 7. 

24. For the positions of the points of contact on the limb of the moon, 

A j immersion ) le f N towardg E ( (l 8 0°-«)- w ) fordireetbnag9m 

I emersion )' 8 ((i80° — *)+«) 

At \ im er . 10 (• , angle from N. towards E. = \ ) l { ~~~ w (• for inverted image. 
( emersion ) ( ( — i) -f- o ) 

At the middle of the eclipse, 

Z cent, shadow from N. towards E. = \ ( 180 ° ~~ 4 ) I for -f dircct , X image. 

t (— i) X ( inverted ) 

To get the same angles from the vertex, the parallactic angle must be deducted 
for the respective times. 






412 



SPHERICAL ASTRONOMY. 



Examples. 



I. ECLIPSE OF THE SUN. 

Let it be required to calculate the circumstances of the solar eclipse of May 15 
1836, as it will be seen at the observatory of Edinburgh. 
The elements of this eclipse are stated at page 362. 

h. m. s. 

Greenwich sidereal time at Greenwich ) „ „ „ 

J. 3 32 58-o 

mean noon J 

Longitude 12 43 • 6 W. 

Edinburgh sidereal time at Greenwich 

mean noon 

Sun's right ascension at <3 . 

Hour angle h at Greenwich mean noon - 
( Greenwich mean time of q . 

(Acceleration 

h at c5 



\> 


20 


i4-4 


«] 




2" / 


•7 


3 


29 
9 


25«2 

IO-8 


/ 




3 


• o3 


- 


83 


• 1 


2 


21 


22*9 
23 «2 


«1 


•/ 




• 8 




84 


• — 



+ 2 i3 



6+21 
/— 6+ 63 



Greenwich mean time of true 6 
T . . . 



60) + 55 (+.87 

5o-4 

"4^6 



h. m. 

2 21 

■ 52 

3 i3 



Constants. 



P — * 

logP 

const. 

logs 



9 . 

a + 



54,23.4 
8.5 

54 i4-9 



3-5i36 7 
9.43537 

2-94904 
1 5' 49" -9 
i8°58'.5 



const. 
c 



A . 
cos I 



sin 5 

6 



9.99902 

4-68555 
4-68457 
3-51254 



8-19711 
9.75001 

7.94712 
4-7172 
+ 9.5121 

+ 2.1764 



Computation for 3 h i3 m , Greenwich time. 

O 1 11 o ' " ' " 

D + 19 33 43 6 + 18 58 29 a + 23 49 

D\ + 9 19 a, + 27 43 

k m. s. 

Edinburgh sidereal time at Greenwich mean noon . . . • 3 20 i4«4 

3 h o m 3 o 29.6 

i3 2.1 

6~33 46-1 



Sidereal equivalent for -J , 



r. 



Moon's right ascension 3 3i 9*0 

A in ■< 



time 
arc . 



3 2 37.1 



+ 45° 39'. 3 



APPENDIX XI. 



413 



m 

cobD 
k 

cos h 
n 

(log • 

1 A«i 



. 7*94712 

. 9.97418 const. . . 5>3r439 

. 7.97294 7.97294 

+ 9-84446 sin A . + 9-8544o 
+ 7.81740 corr. for n . 286 

. 4-7I72 
+ 2-5346 



j log . . + 3.14459 
1 A a . +23' l5" 



+ 5' 42" 



Hi > 
log . 
A A 



+ 9.8544 

+ 2.1764 
+ 2-o3o8 
+ 1' 47" 



h . . +45 3 9 .3 
\ A a + II. 6 

(h) . + 45 5o. 9 

O ' 
0+25 20*9 

D + 19 33.7 
e + D + 44 54-6 



M+3i 54-5 



, + 40 i4-3 



cos . 
cot I 

tan 9 
sin 6 
cos . 

tan (h) . 
tan M . 

cos M . 
tan (e + D) 

tan e 
cos e 

sin c 



logs 

Partial A' . 
Annular A' 



+ 9.8429b 
+ 9-83256 



+ 


9-67552 


+ 
+ 


9-63i56 
9-85oi7 


+ 

+ 
+ 


9-78139 
0-01286 
9.79425 


+ 
+ 


9-92885 , 
9-99864 


+ 
+ 


9.92749 
9-88273 


+ 


9-81022 

8.19711 . 


+ 


8-00733 


• 


2.94904 
444 


. 


2-95348 




i4'58".4 
i5 49 -9 




3o 48 -3 
5i -5 



cos I 

a . 
b . 

check . 



cos e 

B 
const. 



corr. for n x 



A D . 



A. 

A A . 

*1 • 



+ 9.84296 
+ 9»75ooi 

+ 9-59297 

+ 9-8n58 
+ 9.78139 



+ 9.92885 
+ 9-88273 

+ 9-8n58 
+ 5.3i439 

5.12597 

8-19711 

444 

3-32752 

+ 35' 26" 



4-9 19 

+ 1 47 

+ 7T2 



J). . 

Ai> - 
D\ . 

o' C01T. 
S . - 



+ 19 33 43 
+ 35 26 

+ 18 58 17 j 

o (log 
+ 18 58 29 cos D' 

— o 12 y . 



a + 23 49 

A a + 23 l5 

a + O 34 

+ t-53i48 

+ 9-97574 



"hog, 



aj + 27 43 

A a x + 5 42 

+ 22 I 

+ 3-I2090 

- + 9-97M 



+ t .50722 y x 



+ 3-09664 (i) 



414 



SPHERICAL ASTRONOMY 



£ . . + no 28*0 

* . . + 19 53-5 

(S + 1) — i3o 2i«5 

Partial . 

U + 90 4l *I 

a — 221 2-6 

b — 39 4o«4 



V • 

X . 

tan S 

cos S 
W . 

cos . 

n 

log A' 

COS W 

c 
sin a 



+ 1.50722 3/ : 
— 1 -07918 x t 



+ 3.09664 (: 
+ 2-655i4 



— 0-42804 cot 1 + o«44i5o 

— 9.54364 cos t + 9-97328 
+ 1-53554 . . + 1-53554 

— 9-81129 const. 3«5563o 

— 1-34683 + 5-o65i2 (2) 
3.26677 H+ 1.96848 (2) — (1) 

— "8T~o8oo6 . . —8 -08006 



— 3-88842 c — 3-88842 

+ 9-81732 sin 6 — 9>8o5io 



— 3.70574 



+ 3-69352 





Assumed time. 
Beginning . 
Longitude . 
Beginning . 


h. m. s. 
ti — 1 24 39 
3 i3 

1 48 21 

12 44^ 

1 35 3 7 


h. 

t : + I 

. . 3 


m. s. 
22 18 
i3 




Partial. 


Ending 4 35 18^ ™«™* 
& ( mean times 

V. . . 12 44 W. 

Ending 4 22 34 \ Ed,nb " r g h 
b ( mean times 




Annular . 
+ ii5°33'. 9 

— i3o 2i-5 

— 245 55 -4 

— 14 47 -6 


n . 
log A 
cos <a 
c 
sin a 


. — r.34683 
1.71181 


H+ 1 

• —9 

c — 2 

sin b — 9 

+ 1 


96848 
635o2 
33346 
40711 

74057 




u 

~ {$ + 

a 
b 


. — 9>635o2 . 
. —"2T33T46 
. + 9.96047 
— 2.29393 






Assumed time. 
Beginning . . 
Longitude . 
Beginning . 


h. m. s. 
r, — 3 17 
3 i3 


h. 
U + 

. . 3 


m. s. 

55 
i3 




Annular. 


3 9 43 Ending 3 

12 44 W. . . 
2 56 59 Ending 3 


1 3 55 i Nireenwich 
\ mean times. 

12 44 W. 

j Edinburgh 
( mean times, 



Positions of Contacts for direct Image. 



( — — IO *9 
w + 90*7 



(beginning .... no«6) 
Partial contact at ^ end . } 



, , from north towards ] , 
70.8 ) ( east 



Annular contact 



( — — ! 9'9 
u + 1 i5-6 



( beginning • . . . i35«5 ) 
a I ending 95*7 ) 



from : orth towards 



west 
east 



APPENDIX XI. 



416 



For the same angles from vertex we must estimate them towards the east, and 
deduct the angle M, thus 



Beginning — i35 • 5 
M + 3i- 9 



Ending +95*7 
M +3i. 9 



167-4 towards west. 



63-8 towards east. 



Computation for i h 48 m , for an accurate determination of Partial Beginning. 



D + 19 19 35«9 
A + 9 26 



o 



5 7 39.3 



Edinburgh Sid. Time at Greenwich Mean Noon 

rh o m .... 

48 .... 



Sidereal Equivalent for 



I" 



m . 

cos D 
k . 
cos h 
n 



A «i 

h . 

i Ao 



D . 

e + D 



& 's E. A. 

. time 
h in 



7.94712 
9-9748i 
7.97231 
+ 9.95707 
7.92938 
4-7172 



1 — 1 5 a3«2 
i, + 27 38 
h. m. s. 

3 20 i4«4 
1 o 9.9 
48 7-9 
5 8 3 2 . 2 
3 28 18.2 

+ 1 40 i4-o 
+ 25° 3'.! 



4-7172 j log 
+ 2-6466 i a a 



const. . 

sin h . 
corr. for n 
log 



+ 
- o 

+ 25 

+ 



7' 23" 

3.5 

6.9 



-j- 25 io>4 

O ' 

+ 3i 36-7 

+ 19 T 9' 6 
+ 5o 56-3 



M x , . +21 21.1 



+ 48 55.9 



eos . 
cot I 

tan 
sin 
cos . 



tan (A) . 
tan M l 

cos Mi . 



tan 

cos 



sin 
A 



5.3i439 

7.97231 

+ 9-62690 + 9.6269 

370 
+ 2.91730 Q % . . . 2-1764 

+ i3'46"-6 (log. . . +1T80I3 

A A . . + i' 4" 



+ 9'9 5666 +9.95666 

+ 9-83256 cos I . . + 9-75001 



+ 9-78922 
+ 9.71946 
+ 9.79945 

+ 9-92001 
+ 9.67209 
+ 9.59210 

+ 9.96911 . . . . . +0.060H 



G . +9.70667 

B . . . +9.78665 
. check . + 9.92002 



tan(0 + D) +0-09068 coss 



+ 9'8i754 



+ 0-05979 
+ 9-8i 7 54 

+ 9-87733 
+ 8-19711 
•J- 8-07444 



B . 

const. 



corr. for n\ 
log . . . 

AD . , 




+33'3o"-6 



416 



SPHERICAL ASTRONOMY 







log J 
(log 


• • 


2.94904 
5i8 






2*95422 

i5 o-o 


Di . . + 9' 26" 

aA . . + 1 4 






a- 




i5 49-9 


£1 . . . + 8 22 






A' . 




3o 49*9 




D . . + 

Al> . + 


19 19 35-9 

33 3o-6 


a 
A a 


— t 

+ 1' 


5 23-2 

3 46-6 


a i 
A a.\ 


. + 27 38 
• + 7 a3 


D' . . + 

a' corr. . 


18 46 5-3 

2-2 
18 57 39.3 

11 3i«8 

' 
112 39-81 

23 34-49 


llog 

cos D' 

2/ • 
x . 

i tan S 

{sin S . 


2< 

3 
• +9 

— 3 

— 2 

+ ^" 

— 9 
+ 3 


) 9-8 
24299 
•97627 

•21926 
83 99 8 

37928 

96510 


log. 

3/1 • 

Xi . 

cot fi 

COS l i . 


+ 20 i5 

+ 3-08458 
• +9-97627 


a; — 

S . . — 


+ 3-o6o85 (r) 
+ 2 • 70070 

+ o-36oi5 




-f- 9-96215 


--..+ 


25416 


+ 3.254i6 


-(£-K) + 


89 5-32 
89 6-93 


cos 

n . 
log A' 

COS «i . 


+ 8 

+ 3 
+ 8 


20168 


const. 


3-5563o 








45584 
26715 
18869 


6.77261 (2) 


-i . . + 


+ 3.71176 (2) -(l) 

+ 8.18869 


a — 


i-6i 






sin a 


c . 
j sin 1' 


+ 5.52307 

6-463 7 3 

>i — o-2o683 




— 2-19363 












h . 


— o h 2 m 36 s 










Assumed tint 

Beginning 
Long. 


e 1 48 




1 45 24Green»>M.T. 
12 44 W. 



Partial, Beginning . 1 32 4oEdin. M.T 

If the calculation be repeated for the Greenwich time i h 45 m , it will lead to ex 
actly the same result, which is therefore to the accurate second, according to the 
data employed. 

Position of Contact for Direct Image, 



( — ti) — Wl 



■ Mi 



b 
— 23-6 
+ 89.1 



— 112. 7 
+ 21.4 

— i34-i 



The 



point of contact is therefore ]|!j > from \ D °L j- towards 



west 



APPENDIX XI. 417 

H. EQUATIONS FOR REDUCTION OF PARTIAL BEGINNING. 

The data for this computation are taken from the preceding one. 





3«5563o 5 -3 r439 




7.9208 


A' 


3-26715 A . 8-19711 


A « . 


+ 2.9173 


cos 


it 9-96215 corr. for n x 5 18 


sin i) 


+ 9.5198 

■ 




6- 78560 3-5i668 




+ 0-3579 • • e 4" ° 2 *3 


y* 


3-o6o85 a' 3-26715 




90 + i . n3 34-5 


k 


-f-3-72475 b . + o- 24953 


X • Il3 32-2 




k . -{-3.72475 








kb . +3.97428 








D 4-19 19 35.9) 
a corr. 2-2 \ 




a — 2.96530 






cos D' + 9-97627 




6 + 18 57 39.3 




A' y sin xp — 2-94157 




+ 21 58-8 . . 




. A'y eoaxp + 3-12018 




%p —33 32-2 . . . 




j tan xp — 9-82139 
\ cos xp + 9 • 92092 




1 + 23 34-5 






«J"f« — 9 5 7'7 




A'y . . + 3-19926 

a' . . 3-26715 




y o-o32ri . 




. y . . 9-9321 1 




» 77 
*wty + «) 4- 9*99 3 4* 




sin (xp + «) — 9. 238o2 




( + 9.92552 




k . . 3.72475 




(/» 4- 0-84240 




j —2.89488 




h -f-25 3- 5 




\q . . — o h i3 m 5» 




Long. 3 10 .9 W. 




T . . + 1 48 




H + 28 14 -4 




Z* . . + 2 i 5 




cos i) 4~9*9748i 




cos Z> +9-97481 




cos c -f- 9-96215 




sin « + 9-60200 




b . -f 0-24953 




&6 . +3.97428 




L . -f- 0-18649 




L" . + 3.55109 




sin D 4~9*5i977 




sin D + 9. 5 1 977 




cos * -f- 9.96215 




sin i + 9-60200 




+ 9.48192 




+ 9-12177 




cos x — 9«6or34 




sin x +9-96228 


*'- 


^sm . — 9.90109 Y 


-H+ 


s,° 47'- r an - ±°- 84 °' 1 

' ^s:n . + 9-99552 




#+28 i4«4 -(-9.70025 


H+ 


28 i4-4 +9-96676 


f 


. — 24 3a .4 6 . 0.24953 -y 


+ 


no r -5 kb . 3.97428 




y' . 4-9*94978 




y" . + 3'94io4 




27 







418 SPHERICAL ASTRONOMY. 

"We have hence, for the Greenwich time t of beginning, at any place whose lat- 
itude is I, = north, — south, and longitude A, + east, — west, the two following 
equations, which may be safely depended on for any place in Scotland or the North 
of England. 

cos w =o -8424o— [0-18649] sin I -f- [9 • 94978] cos / cos (A— 24 32'. 4) 
t=2 h i m 5»— [3.72475] sin w + [3 .55109] sin/ — [3.94104] cos I cos (A + 110 i'«5) 

Contact on ©'s limb, u> + 23° 34' -5 from the north towards the west. 
As a check on this calculation take the assumed radical place, Edinburgh, and 
l = -\-55° 46'- 9, A = — 3° 10' -9, giving o> = 89° 6' -9 and t = i 1 ' 45 ra 24 s , which 
perfectly coincide with the results of the original calculation. 

Similar calculations for the ending of the eclipse give the equations, 

cos w=o. 93848 — [0-20291] sin 1+ [9.88677] cos lcos(\ + 27 6'«7) 
*=i h 38 m 33 s + [3.6689o]sin u + [3-35544]sin/— [3.90073] cos Z cos (a + 1 53° 3'- 8) 
Contact on O's limb, w — 16 56' • 2 from the north towards the east. 
Also by calculating with T= 3 h i3 ra for the annular phase there will result 

cos 01 = 29.66600— [1-75 1 59] sin l+[i -46950] cos I cos (A+ i° 42' «4) 
«=i h 43 m 7 3 T [2 • i4475] sin u> + [3-45484] sin l-[3 .92550] cos I cos (A + i3i° 55' .9) 

Contact on O's limb, — 19 53' -5 T w from the north towards the east, 
the upper sign appertaining to the beginning and the under sign to the ending. 
If cos u» > 1, the place will be without the limits, and the eclipse will not be annular. 
By taking I = + 55° 46' -9, A = — 3° 10' .9, the results will exactly correspond 
with the special calculation. 

Note. — The expression of cos w for the annular phase, as the appearance of this 
phase is comprised within narrow limits on the surface of the earth, will afford a 
very convenient and simple determination of the places which range in those lim- 
its as well as those which range in the central line ; and we may expect very ac- 
curate results throughout the portion of country originally taken into considera- 
tion. Thus for the southern limit we must obviously have cos &> = + 1, for the 
central line cos 10=0, and for the northern limit cos &> = — 1 ; and hence the 
following conditions: 

i+ l ) ( southern limit, 

o > for < central eclipse. 



( northern limit. 



By making the assumptions 

ri cos A 77 = y' cos (A -\- ip') ) 

ri sin N' = U \ (r) 

they will give 

C — p + 1 } ( southern limit } 

ri cos (JV + I) = < — p > for ■< central eclipse > ....(*) 

( — p — 1) ( northern limit ) 

If we therefore take any meridian whose east longitude is A, these two equa- 
tions (r), («) will serve to determine the extreme latitudes I, on this meridian, be- 
tween which the eclipse will be annular as well as that where it will be central. 
For the preceding eclipse, these equations will be 

ri cos A 77 = [1 .46950] cos (A -f i° 42' 4), 
ri sin A 77 = [1 -75159] ; 

r — [1.45737]} C southern limit 

ri cos (A 7 " + I) — \ — [1 .47226] > for < central eclipse. 

( — [1 . 48665] ) ( northern limit. 



APPENDIX XI. 



419 



If we take, for example, the meridian of Edinburgh, and use A = — 3° io'-o, 

there will result, 

o ' 

Extreme southern point of annular appearance, N. 54 19-7 

Point of central appearance, N. 55 20 -4 

Extreme northern point of annular appearance, N. 56 21*7 

which are geocentric latitudes. 

III. CALCULATION OF THE TRANSIT OF MERCURY, 
November 1, 1835. 
The conjunction in right ascension takes place a^out 7 h 38 m ; take therefore 
T= 7 h 4o m , and we readily find from the ephemeris the following data: 

S — 16 i5' 58". 2 



D — 16 22 4*2 

A— 2 32-6 

« 4-8 

P 12.66 



« -f o 10.95 
<*i + 5 32.7 
v 16 10-4 
tt 8-66 



With these quantities, the calculation, for external contact of limbs, is as follows? 

P 



a- 16 IO-4 

« 4-8 

A 16 l5-2 



a + I • o394l 

cos 8 + 9-98226 
cos 8 + 1-02167 



12-66 
8-66 
4-oo 



; '1 

•o ) 



+ 




6 


6. 










# 






* + 


I 


38 


•7 


• 


( — 


25 


32 


.3 





i — 16 i5 58-2 
3 corr. 

D — 16 22 4*2 a COS 8 + I -02(67 

+ 2-56348 
tan \p + 8-45819 
cos U/ + 9-99982 

+ 2-56366 
A £ • 98909 
y + 9-57457 
y+ 9.57457 £+3.99643 

cos (^ + «) + 9*96109 sin (^ + 1) — 9-60749 
cos w + 9. 53566 
o ' 

w + 69 55-4 



. o- 60206 
A 2-98909 
b + 7-61297 

«! + 2-52205 

+ 9.98226 
a, cos S + 2-5o43r 

n 1 ~ 2 i'2. 6 . 



. . — 2-I835S 
cos 8 + 2-5o43r 



« — 25 32-3 



+ « — 23 53-6 



sin w + 9.97278 
k + 3.99643 

k sin w + 3.969^1 





— 3.1 


7849 


<f- 


Ji. in. 
•O 25 


s. 

8.3 


T + 


7 4o 




q + 


8 5 


8.3 


. 


2 35 


i5.6 



( tan 
-3 \ 


— 


9.07924 


{ cos 


+ 


9 . 9 5535 


A 




2-98909 


const 




3-5563o 




+ 


6- 50074 


k 


+ 


3 • 99643 


b 


+ 


7-61297 


kb 


+ 


1 .60940 


sin w 


+ 


9.97278 


k" 


+ 


1.63662 


cos 8 


+ 


9 98226 


k" cos 8 


+ 


1. 6 1 888 



Mean time 



ime of j 



ingress 5 29 52-7 
egress 1® 4o 



52-7? 
23-Q ) 



for the centre of the earth. 



420 SPHERICAL ASTRONOMY. 

Constants for Reduction of Ingress. 



^in 



Equa. 


+ 

+ 
+ 
+ 


h. m. 
5 29 

16 


s. 

52 
10 


•7 
• 


| time 


5 46 


2 


•7 


( arc 


86° 


3o' 


■7 


— < 
w 


25 

69 


32 

55 


• 3 

4 


i — w 


44 


23 


• 1 



'AT in J 



. . cos + 9-854ro . . . sin — 9.84477 
sin S — 9.44733 

— 9-3oi43 .... —9.30143 

t" — # — ro5 58 -3 j tan + 0.54334 

,// i : 7~7 (sin — 9.98200 

<//' — 1927-6 y l_ f_ 

+ 9.86187 

&" cos 5 + i» 61 888 k" + 1 -63662 

L" + 1 .47298 y " + 1 .49849 

Constants for Reduction of Egress. 
b. m. s. 
10 4o 23.9 
Equa. + 16 9- 2 

time + 10 56 33- 1 



i64° 8'. 3 



— 1 + w + 


95 27 .7 . 


. cos — 8*97854 . 
sin 6 — 9.44733 


. . 810 + 9.99802 






+ 8-42587 . 


. . . > 8-42587 




88 28 .0 . 




j tan + 1 .57215 


252 36 .3 

107 23 .7 


k" cos 6 + 1 -61888 


{sin + 9.99984 

+ 9-99818 
k' 1 + 1 '63662 






L" — 0.59742 


y" + l'6348o 



The former part of the calculation repeated for the times 5 h 3o m and io h 4o m we 
shall find more accurate times of ingress and egress, for the centre of the earth, 
to be 5 h 29™ 56 s and io h 4o m 3i s , which, however, still cannot be depended on 
within a few seconds. More reliance can be placed in the amount of reduction for 
parallax. The times reduced for any place whose north latitude is /, and east 
longitude X, viz. : 

Ingress, Nov. 7 fl 5 h 29™ 56 s + [1 '473a] p sin / — [t -4985] p cos I cos (A — 19 28') 
Egress, " * 10 4o 3i +[0-5974] p sin / + [i-6348] p cos I cos (A — 107 24') 
will indicate, with considerable accuracy, the difference between the times at any 
two places. 

The positions of the contacts on the sun's limb, for an inverted image, will be 

„ ( ingress 44° 23' ) l west. 

Contact at i ^ e i0 r from the north towards the ■{ 

( egress ..... 90 20 ) ' 



APPENDIX XI. 



421 



IV. OCCULT ATION OF A STAR. 

On January 7, 1836, the star t Leonis, whose right ascension ia io h 23 m 26*'4 and 
declination N. i4° 58' 39", will be occulted by the moon. 

Limits of Latitude. 
At the time of true <3 in right ascension, viz., I2 h i2 m 17 s , we have the follow- 
ing data: 

O 4 44 4 14 

D + i5 33 2 
t + 14 58 39 
D — <* + o 34 23 
with which we proceed thus : 



Z>i — 11 47 




a x + 3o 4i 




P + 56 4 




n 
~P~~ 


•5699 


const, -f 


•2725 



A— 11 47 • —2.84942 
a t + 3o 4i • -f- 3'265o5 

— 9-58437 o 

6 + i4°59' cos + 9.98498 w x + i47 2 4 • natcos - 

nat. cos — 



I tan — 9.59939 w 2 + 107 18 

« — 2I 4l 1 — 7TT t + 21 41 

( cos + 9.96813 

diff. dec + 34' 23" . + 3-3i45o 



+ 86 3 7 



P + 56' 4" 
p- + -5699 



» + 3.28263 
+ 3.52686 



29 74 

log. cos -j- 9.9681 (1) 
log. cos + 8-8833 (2) 
log. cos <S -f- 9.9850 (3) 



+ 9-75577 



8 + 63 5i . log.cos -f 9-9531 (i) + (3) 

J, + 83 33 

/ 2 — 4 i4 — log. sin /„ + 8 • 8683 (2) + (3) 



The star may therefore be occulted between the parallels of latitude N. 83° 33' 
and S. 4° i4'- The parallel of Greenwich is within these limits; and if the hour 
angle of the star be computed roughly for the meridian of Greenwich, the star will 
be found to be considerably elevated above the horizon. A special calculation for 
the observatory of Greenwich will consequently serve as an example of the cir- 
cumstances for a particular place. 

Calculation for Greenwich Observatory. 
Constants <f>' l \ <p' 2 \ 0( 3) . 
p . 9.99913 

+9-79610 
+ 9.79523 . . . 
+ 9»9o38i const . 



cos I 

cot I 

'0 



+ 9.89142 # 



(») 



+ 9.79523 
9-41916 
-f 9-21439 



These will be constant for all occupations at Greenwich. 



b. m. s. 
19 4 22.4 
10 23 26.4 



Sidereal time at mean noon . 
Star's right ascension 

h at mean noon . . . . — 1 5 1 9 4 • o . . 

Mean time of true (5 . . + 12 12 T . 

Acceleration -f 2 acceleration 

h at true 6 • • — 3 5 ( time 



h. m. s. 

— i5 19 4.0 

+ 116 

+ 1 49-4 

— 4 ir i4-6 



— 62°4S'-7 

With this and a x = 3o'-7 we find, by the table at p. 405, T=z n h 6 m . 
h at mean noon is put down negatively, in order to have more readily the other 
values of h less than I2 h or 180°. 



422 



SPHEKICAL ASTRONOMY. 



P 56' 4" 


+ 3.52686 


. 


+ 3.52686 . . 


. +3.52686 


f w. . 


+ 9.89142 


cos h 


+ 9-65983 sin A 


— 9-94915 




+ 3.41828 




+ 3.18669 


— 3..47601 


cos a 


+ 9.98499 


sin b . 


+ 9-4i236 sin i 


. +9«4i236 




| 4-3-4o32 7 




+ 2 • 59905 


— 2.8883 7 




1+ 42' 11" 


fP> . 


+ 9.79523 £' 3 ) 


. +9.21439 




+ 48 


. . . 


+ 2.39428 


( — 2.10276 




+ 38 3 






I- »'7» 


D-i . 


+ 47 22 




A 


. II 42 


a- 


+ 9 19 




Xi 


. — 9 35 


r ' 


— 33 54" 
— 3-3o835 




r 


. + 3o'44" 
= 3.26576 


cos 6 


+ 9.98499 
j —3.29334 

\— 32' 45" 




cos £ 


. +9.98499 
j +3-25o 7 5 
I + 29' 4i" 


P sin A 


— 3.47601 


P . 


. 3.52686 Pcos 


h +3.18669 


*0>. . . 


+ 9.79523 


eonst. 


. 9.43537 0(3) . 


+ 9.21439 




j —3-27124 
1- 3i' 7" 


A' . 


2-96223 


j + 2.40108 








\+ 4' 12" 


r •■ 


— 1 '38" 




f" 


. + 25 29 


— 1 «99r23 


+ 3-i844i 


X . . . 


+ 2. 7 474i 




Xi 


. —2-75967 


tan £ . . 


— 9.24382 


8 . . 


— 9 56« 6 cot 1 


— 0.42474 


cos £ . 


+ 9-99343 


* . 


— 20 36-6 cos 1 


. +9.97128 


JF . . . 


+ 2.75398 




. . . . W 


. +2.75398 


cos — (S + 


t ) + 9-9 3 5o8- 


-(£+«) + 3o 33-2 


3-5563o 


n . 


+ 2.68906 






+ 6-28i56 


A' . . . 


2.96223 




H 


. +3-09715 


COS w 


+ 9.72683 


. b> 


+ 57 47 rp . COS o> 


+ 9-72683 


< 


+ 3.37032 


a 


— 27 1 3> 8 c . 


. + 3.37o32 


&in a . 


— 9-66045 

— 3-o3o77 


b . 


+ 88 20 • 2 sin b 


• +9'999« 2 
+ 3.37014 


*i . . . 


— o h i7 m .9 




fe . . . . 


+ o h 39 m .i 


T . . . 


11 6 

10 48 . 1 




T. . . . 
. Emersion 


11 6 


Immersion 


11 45 1 mean times. 


Acceleration i -8 




Acceleration 


2 >o 


S. T. mean noon 19 4-4 




S. T. mean noon 


19 4 -4 


Immersion 


5 54-3 


. 


. Emersion 


6 5 1 .5 . sid. times. 


Star's R. A. 


10 23 .4 




Star's R. A. . 


10 23 «4 


c Im. A . 


— 4 29. 1 = 


= -6 7 ° 


j Em. /* = . 
( Parallactic Z. 


— 3 3i .9 = — 53* 


( Parallactic/ —39° .7 


— 36°. 9 


(-') • • 


. + 20 «6 




(-<)• • • 


+ 20 -6 


> 


. +57 8 




a> . . . 


+ 5 7 -8 


From \ north 

( vertex + 2 - 5 ) 


to the east. From \ 


t 78 ^Itotheea.t 




( vertex + iid «3 ) 



APPENDIX XI. 423 

< "Tiese angles are for the inverted image ; and, being estimated towards the east, 
* negative values must be considered as towards the west. The declination of 
> he star gives for the latitude of Greenwich a semi-diurnal arc of j h 23 m ; as this 
exceeds the value of h both at immersion and emersion, the immersion and emer- 
sion will both occur above the horizon. 

V. CALCULATION OF THE ECLIPSE OF THE MOON, 

April 30, 1836. 
The opposition or full mocn takes place at 19 11 58 m . For the computation assume 
the time 20 b o m . 

i9 h 20 h 2I h 

h. m. s. h. m. s. h. m. 8. 

J'sR. A. . . i4 32 5i«35 . . 14 35 11*19 • • i4 37 3i*43 
0'sRA. + n b i4 33 52*38 . . 14 34 1*91 . . 14 34 11 -45 



H 



time — 1 i*o3 -f- 1 9- 28 +3 19*98 

space — i5' i5" + 17' 19" + 5o' o" 



*" + 32' 34" 



a = 


+ \7 <9 + 32 
-f 5o T 


. ai = + 32' 38" 
4i ' 






i9 h 


20 h 


21** 




' " 


' " 


O ' " 


J> 's dec. . . 


, — 14 519). 


. — i4 19 58) . . 


— i4 34 32 ) 
5? 


a cor. . . 


0's dec. . . 


+ i5 6 35 . 


. -j- 1 5 7 20 . 


+ i5 8 6 


x . . . 


4- 1 1 16 


+ 47 21 


+ 33 29 



S 


+ 19° 


3o' 


•7 


t 


— 23 


43 


•9 


0+0 + 4 


i3 


• 2 . 






External 


id 


+ 35 


38 


•7 


a 


-3i 


25 


*5 


6 


-i-3 9 


5i 


•9 



+ 61 ' l6 " _ i3' 55" 
x = + 47 21 __ l3 52 *i = — r3' 54'" 
+ 33 29 

a-|-3*oi662 ai + 3-29i8i P — Qo' 19" 
cos -0 + 9*98627 . . 4-9*98627 

P 3*55859 

9'999 2 9 

3.55788 
tan # + 9*54942 cott — 0*35691 . 

cos S 4-9 *9743 1 cos i-|- 9*96163 ' I 

W+ 3.47916 . . +3.47916 <r . r5 53 



*/ + 3*00289 2/1 + 3*27808 

X + 3 • 45347 Xi — 2'()211J 



cos +9.99882 3*5563o 44 29 

n + 3 * 47798 +6*997^9 ^ J A A 

. . A' 3*568o8 ~ 45 1 3 shadow 

#+3.71901 H 

. 7 s 16 26 

cos <•> + 9*90990 . . +9*90990 

1 o o , V~o » f ( 61 39 external 

cf 3 -8091 1 c + 3*8oorr A \ / . , 

7 ,. . , _ 7 . ( 28 47 internal 

sin a — 9*71716 siu6 + 9*8o685 

— 3.52627 +3*61596 

t-i — 56 m .o * a + I** 8 m .8 

Assumed time; 20 11 o ... 20 o 



Beginning 19 4 *o Ending 21 8 «8 Green \v h mean times. 



424 



SPHERICAL ASTROKOMT. 



For the times at any other place, it will only be necessary to take into account 
the difference of longitude* 

The positions of the points of contact on the limb of the moon may be deter- 
mined in the same manner as those of an occultation, and will here be unnecessary. 

As A' for internal contact with shadow is less than n, no internal contact can 
take place, and therefore the eclipse is only partial. 

The contacts with the penumbra are to be determined in a similar manner from 
the same values of n, H, and will also be unnecessary here. 

i\' for external contact with shadow 61' 39'' 
n 5o 6 



11 33 



.Eclipsed . 

which divided by 2 s = 32' 52", gives o«35i for the magnitude of the eclipse, the 
moon's diameter being unity. 



APPENDIX XII. 



EQUATION OF EQUAL ALTITUDES. 



Let P be the pole, Z the zenith, S r 
the place of the sun in the afternoon, S 
the place he would have occupied had his 
declination or polar distance P S remain- 
ed unchanged. Make 

' I = latitude of place = 90° - P Z 

x = declination of sun = 90° — P S 

a = altitude of sun = 90° — Z S 
P = hour angle Z P S ; 

then in the triangle Z P S r 

sin a = sin I sin x -f- cos I cos x . cos P 

Differentiating, supposing x and P alone to vary, we have 

d x . cos x . sin I = cos I . cos x . sin P . d P + cos I . cos P . sin x d x yi 

or % 

d x . (cos x . sir I — cos I . cos P . sin x) = d P . sin P c6s I cos x ; 

whence 

dP = dI A^i-^i\. ..... ( i) 

\sin P tan PI v ' 




(«) 



APPENDIX XIII. 



425 



Denote by 8 the change in declination from the next preceding to the 
next following noon or change in 48 hours, and by t the interval in hours 
between the epochs of equal altitudes in the morning and afternoon. Then 

48 : 8 : : t : d x, 
whence 

S.t 

48 ' 



dx 



also 



*-? = **. 



which substituted in Eq. (b) give 

, „ 8 . tan I . t 
d P = 



8 . tan x . t 



48 . sin. 1±t 48 . tan 1\ t ' 

converting both members into time and taking one half, we have, after 
writing d for a?, and making l y = | X n • ^ = To ^ ^i 
S . tan I . t 8 . tan d . t 



1440 . sin H I 1440 . tan 7-J t 



as in the text, page 187, 



APPENDIX XIII. 



w 



CORRECTION FOR DIFFERENCES OF REFRACTION. 

Let P be the pole, Z the zenith, and 5 
the place of the sun had the air undergone no 
change, and S' the place as determined by 
a change of atmospheric refraction. Then, 
employing the same notation as in the pre- 
ceding appendix and resuming its equation 
(a), regarding the altitude a as referring to 
the place S and P to the hour angle 
Z P S, we have, writing d for re, 

sin a = sin I . sin d + cos I . cos d . cos P, 

and denoting the altitude of S' by a' and the hour angle Z P S' by P f 

sin a' = sin I . sin c? -}- cos I . cos c? . cos P' ; 

and by subtraction, 

sin a' — sin a = cos J . cos c? (cos P / — cos P) ; 




426 

but 



SPHERICAL ASTRONOMY. 



sin a' — sin a = 2 sin i (a' — a) . cos i (a' + a), 
cosP' - cosP = 2sin-i (P - P') . sin i (P' + P) ; 

whence by substitution, 

sin i (a'— a) . cos i (a'+ a) = cos Z .cos d . sin 1 (P—P f ) . sin A (P' + P), 

and because a and a' as also P' and P differ by very small quantities, the 
above becomes, by transposing and dividing, 

p _ p> = ( q/ ~ a ) • cos a 

cos / . cos d . sin P" 

But denoting the refraction in the afternoon by r' and that in the morning 

by r, we have 

a' — a = r' — r ; 

substituting and converting both members into time, and writing i u for the 
first member, we have 

j (/ — r) . cos a 

1,1 = TT ' cos/.cosrf.sinP' 
as in the text at page 188. 













J. 





V 



TABLES. 



TABLE L 



Mr. Ivory s Mean Refractions ; with the Logarithms and their Differ- 
ences annexed. 



■ 

| Zenith 
Dist. 


Mean 
Effraction. 


Log. 


Diff. 


Zenith 
Dist. 


Mean 
Kefraction. 


Log. 


Diff. 


o 

I 


O I «02 


o-oo85 


3012 



25 


O 27-24 


1-4352 


201 


2 


2-o4 


0.3097 


i 7 63 


26 


28.49 


1-4547 


i 9 5 


3 
4 
5 


3-o6 
4-o8 
5- ii 


o-486o 
0.6112 
0.7086 


1 25a 

974 
796 
6 7 5 


27 
28 
29 


29.76 
3i «o5 
32-38 


i.4 7 36 
1. 4921 

I -5l02 


189 

i85 
181 


6 


6-i4 


0.7882 


3o 


33.72 


I .5279 


177 
i 7 3 


7 


7.17 


o.8557 


58 7 


3i 


35.09 


1.5452 


8 
9 


8.21 
9*25 


0.9144 
. 9663 


519 

466 


32 

33 


36-49 
37.93 


I .5622 

1.5790 


170 
168 


10 

ii 

12 


io«3o 
n-35 

12-42 


1 -0129 
i-o553 
1-0941 


424 
388 
35 9 
334 
3i3 


34 

35 

• 36 


39.39 
40.89 
42.42 


1.5954 
1.6116 

I .6276 


164 
162 
160 


i3 

i4 


i3-49 
14-56 


1 .i3oo 
i.i634 


37 
38 


44- 00 
45.6i 


1-6435 
1 .6591 


1 5 9 
1 56 


i5 


15-66 


I-I947 


294 
278 
265 


3 9 


47-27 


1.6746 


i55 


16 

17 


i6. 7 5 
17-86 


I -224l 
I «25l9 


4o 
4i 


48.99 
5o-75 


1 .6901 

1 -7o55 


i55 
1 54 


18 


18.98 


I-2 7 84 


252 


42 


52.57 


1 .720-7 


i5a 


J 9 


20. r 1 


i.3o36 


24l 


43 


54-43 


1-7358 


i5i 

l52 


20 


21 -26 


1.3277 


230 


44 


56-35 


1 -75io 


21 


22.42 


i.35o 7 


222 


45 


58-36 


1 -76611 


i5i 


22 


23-6o 


1.3729 


2l5 

207 


46 


1 c43 


1. 78123 


l5l2 


23 
24 


24-80 
26-01 


i.3 9 44 
i-4i5i 


4i 
48 


2.57 
1 4-8o 


1.79637 
i.8ii55 


i5i4 
i5i8 



428 



SPHERICAL ASTRONOMY, 
Table I. — (Continued.) 



Zenith 
Dist. 


Mean 
Eefraction. 


Log. 


Diff. 


Zenith 
Dist. 


Mean 

Eefraction. 


Log. 


Diff. 


o ' 
49 O 


1 7«il 


1.82678 


i523 


' 
72 3o 


3 3-23 


2-26299 


429 


5o 


9-52 


I-84208 


i53o 


4o 


5-o6 


2-26732 


433 


5i 


12-02 


1-85747 


1539 


5o 


6- 9 3 


2-27168 


436 


52 


i4-64 


1-87298 


i55i 


73 00 


8-83 


2-27608 


44o 


53 


i 7 .38 


1-88863 


1 565 


10 


10-77 


2-28o5l 


443 


54 


20-24 


1 • 90440 


1577 


20 


12-74 


2 • 28498 


447 


55 


23-25 


1 .92036 


1596 


3o 


14-75 


2 • 28948 


45a 


56 


26-41 


1-93653 


1617 


4o 


16.80 


2 • 29402 


454 


57 


29.73 


1-95291 


1 638 


5o 


18.88 


2-29860 


458 


58 


33-23 


1 .96955 


1664 


74 00 


21 -OI 


2-3o322 


462 


59 


36- 9 3 


1.98646 


1691 


10 


23-l8 


2-30789 


46 7 


60 


4o-85 


2«oo368 


1722 


20 


25-39 


2-31259 


470 


61 


45-oi 


2.02124 


1756 


3o 


27-66 


2.31734 


475 


62 


49-44 


2*03918 


1794 


4o 


29.95 


2-32213 


479 


63 


54-17 


2-05754 


i836 


5o 


32>3o 


2-32696 


483 


64 


59.22 


2.07635 


1881 


75 00 


34.70 


2-33i84 


488 


65 


2 4-65 


2*09567 


1932 


10 


3 7 -i6 


2.33677 


493 


66 


io- 48 


2. n555 


1988 


20 


3 9 - 65 


2-34174 


497 


67 


16-78 


2«i36o3 


2048 


3o 


42-21 


2.34676 


5o2 


68 


23-6i 


2-15719 


2116 


4o 


44-82 


2-35i83 


507 


69 


3 1 «o4 


2*17910 


2191 


5o 


47-48 


2-356 9 5 


5l2 


70 00 


39-16 


2«20l85 


2275 


76 00 


59-21 


2-36212 


5i 7 


10 


40-59 


2.20573 


388 


10 


53«oo 


2-36 7 35 


523 


20 


42-04 


2-20963 


390 


20 


55-85 


2.37263 


528 


3o 


43-52 


2-2i356 


393 


3o 


58- 76 


2.37796 


533 


4o 


,45-02 


2-21752 


396 


4o 


4 i-74 


2-38334 


538 


5o 


46-53 


2-22l5o 


398 


5o 


4-79 


2.38879 


545 


71 00 


48- 08 


2-22552 


402 


77 00 


7.91 


2-39430 


55i 


10 


49-65 


2-22956 


4o4 


10 


ii-ii 


2-39987 


557 


20 


5i-25 


2-23363 


407 


20 


14-39 


2-4o55o 


563 


3o 


52-8 7 


2.23773 


4io 


3o 


17.74 


2«4ii 19 


569 


4o 


54-53 


2.24186 


4i3 


4o 


21 .19 


2.41695 


576 


5o 


56-21 


2 • 24603 


417 


5o 


24.72 


2.42278 


583 


72 00 


57.92 


2-25o22 


419 


78 00 


28.33 


2.42867 


589 


10 


59-66 


2.25445 


423 


10 


32. 04 


2-43463 


596 


20 


3 i-43 


2-25870 


425 


20 


4 35-84 


2 -44o66 


6o3 



TABLES. 
Table I. — (Continued.) 



429 



Zenith 
Dist. 


Mean 
Eefraction 


Log. 


Diff. 


Zenith 
Dist. 


Mean 
Refraction. 


Log. 


Diff. 


O ' 

78 3o 


4 3 9 . 7 5 


2.44677 


611 


' 

84 20 


8 55-25 


2 -72856 


1069 


4o 


43.76 


2.45295 


618 


3o 


9 8-88 


2-73943 


1092 


5o 


47-88 


2.45921 


626 


40 


23.16 


2-75o63 


iii5 


70 00 


52.12 


2-46556 


635 


5o 


38-12 


2.76202 


1 1 39 


10 


56-47 


2.47198 


642 


85 00 


53-84 


2-77367 


ii65 


20 


5 0.94 


2.47848 


65o 


10 


ic io«35 


2. 7 8558 


1 1 91 


3o 
4o 


5.54 
10.28 


2-48507 
2.49176 


659 
669 


20 
3o 


27.73 
46- o3 


2.79777 

2«8l025 


1219 

1248 


5o 


i5-i6 


2.49853 


677 


4o 


11 5«3o 


2-82302 


1277 


80 00 


20- 19 


2-5o54i 


688 


5o 


25-66 


2.836i 1 


1 309 


10 


25-36 


2-5i237 


696 


86 00 


47-1.5 


2.84951 


1 34o 


20 


30.70 


2.51944 


707 


10 


12 9.88 


2-86325 


i3 7 4 


3o 


36- 20 


2.52660 


716 


00 


33.97 


2.87735 


i4io 


4o 


4i-88 


2-53387 


727 


3o 


5 9 - 5i 


2.89182 


1447 


5o 


47*74 


2.54i25 


738 


4o 


i3 26.61 


2 • 90666 


1 484 


81 00 


53.79 


2.54874 


749 


5o 


i3 55.4o 


2-92189 


i523 


10 


6 o«o4 


2.55635 


761 


87 00 


i4 26.04 


2.93754 


1 565 


20 


6-5o 


2 • 56407 


772 


10 


14 58. 7 i 


2-95362 


1608 


3o 


i3-i8 


2.57192 


785 


20 


1 5 33.6o 


2-97016 


1 654 


4o 
5o 


20-09 
2706 


2.57989 
2.58800 


797 
811 


3o 
4o 


16 10.89 
16 5o-8 


2-98717 
3 • 00466 


1701 
1749 


82 00 


34-68 


2.59624 


824 


5o 


17 33-6 


3.02267 


1801 


10 
20 


42-3 7 
5o-33 


2.60462 
2«6i3i3 


838 
85i 


88 00 
10 


18 19.6 

19 9.0 


3«o4l22 

3«o6o3i 


i855 
1909 


3o 


58- 5 9 


2-62179 


866 


20 


20 2.2 


3 • 07998 


1967 


4o 


7 7-19 


2-63o62 


883 


3o 


20 59.6 


3.10024 


2026 


5o 


i6-i3 


2.63961 


899 


4o 


22 1 .7 


3«i2ii3 


2089 


83 00 
10 


25 -4o 
35-o5 


2.648 7 5 
2-658o6 


914 
93i 


5o 
89 00 


23 8.9 

24 21.8 


3.14268 
3.16489 


2i55 
2221 


20 
3o 


45.io 
55-58 


2.66755 
2.67722 


949 

967 


10 
20 


25 40.9 

27 7-i 


3.18779 
3«2ii4o 


2290 
2 36i 


4o 

5o 

84 00 


8 6-5o 
17-90 
29.80 


2-68708 
2.69714 
2.70740 


986 
1006 
1026 


3o 
4o 
5o 


28 4o-8 

3o 23-2 

32 i5-o 


3-23574 
3.26083 
3.28667 


2434 
2509 
2584 


10 


8 42.24 


2.71787 


io Ai 


90 00 


34 i 7 -5 


3- 5 1 334 


2667 



430 



SPHERICAL ASTRONOMY. 



TABLE II. 
Mr. Ivory's Refractions continued : showing the logarithms of the correc- 
tions, on account of the state of the Thermometer and Barometer. 



Thermometer. 


Barometer. 




Logarithm. 




Logarithm. 




Logarithm. 


o 
80 


9.97287 



5o 


O- OOOOO 


in. 
3i«o 


o-Ol424 


79 


9.97326 


49 


O • 00094 


3o«9 


0-01248 


78 


9.97416 


48 


0-00190 


8 


o-on43 


77 


9 «975o6 


47 


o. 00285 


7 


0-01002 


76 


9.97596 


46 


o-oo38o 


6 


• 00860 


75 


9.97686 


45 


0-00476 


5 


0-00718 


74 


9-97777 


44 


o- 00572 


4 


0-00575 


73 


9.97867 


43 


• 00668 


3 


0-00432 


72 


9.97958 


42 


• 00764 / 


2 


0-00289 


7i 


9.98049 


4i 


o- 00861 


1 


o«ooi45 


70 


9.98140 


4o 


0-00957 


3o-o 


0.00000 


69 


9.98231 


3 9 


o-oio53 


29.9 


9.99855 


68 


9.98323 


38 


o-oii5i 


8 


9.99709 


67 


9.98414 


37 


0-01248 


7 


9-99563 


66 


9.98506 


36 


o-oi346 


6 


9.99417 


65 


9.98598 


35 


o- 01 444 


5 


9.99270 


64 


9.98690 


34 


o-oi54x 


• 4 


9.99123 


63 


9.98783 


33 


o«oi64o 


3 


9.98975 


62 


9.98875 


32 


o- or 738 


2 


9.98826 


61 


9.98969 


3i 


0-01837 


1 


9.98677 


60 


9.99061 


3o 


0-01935 


29.0 


9.98528 


5 9 


9.99154 


29 


o«o2o33 


28-9 


9.98378 


58 


9.99248 


28 


o-o2i33 


8 


9.98227 


5 7 


9.99341 


27 


0-02232 


7 


9.98076 


56 


9.99434 


26 


- «o233i 


6 


9.97924 


55 


9.99529 


25 


0*02432 


5 


9.97772 


54 


9.99623 


24 


o«o253i 


4 


9.97620 


53 


9.99717 


23 


o«o263o 


3 


9.97466 


52 


9 • 998 r 1 


22 


0-02730 


2 


9-973i3 


5i 


9.99906 


21 


0-02832 


1 


9. 97 i58 


5o 


o« 00000 


20 


0*02933 


28.0 


9.97004 



r 



p* 



r 



TABLES. 



m 



TABLE III. 

Mr. Ivor tf s Refractions continued: showing the further quantities by 
•which the refraction at low altitudes is to be corrected, on account of 
the state of the Thermometer and Barometer. 



Zenith 
Distance. 


T 


B 


Zenith 
Distance. 


T 


B 


O ' 

75 O 


— 0.009 




' 
86 3o 


— o«3i7 


+ o-5i 


76 


COI2 




86 40 


0.345 


o.56 


77 


o«oi5 




86 5o 


0-376 


0.62 


78 


0.018 




87 


o-4io 


0.68 


79 ° 


0-023 


„ 


87 10 


0.448 


0.75 


80 


o«o3o 


+ o«o4 


87 20 


0.490 


o-83 


81 


0"040 


o-o5 


87 3o 


0-538 


0.91 


81 3o 


o-o46 


0-07 


87 4o 


0.593 


I "OI 


82 


o«o53 


0-08 


87 5o 


0-654 


li3 


82 3o 


o-o63 


O-IO 


88 


0-722 


1.26 


83 


0-074 


O'll 


88 10 


0.799 


i-4i 


83 3o 


0-089 


o-i3 


88 20 


0-887 


1.59 


84 


0-107 


0.16 


88 3o 


0-987 


1.79 


84 3o 


o- t3o 


0'20 


88 4o 


I 'IOI 


2. 02 


85 


0-159 


0«25 


88 5o 


i. 2 3i 


2-29 


85 10 


0.171 


0.26 


89 


i.38o 


2.61 


85 20 


0.184 


0.28 


89 10 


i-55i 


2.98 


85 3o 


0.198 


o-3i 


89 20 


L749 


3-4t 


85 4o 


0«2l3 


0.33 


89 3o 


1.977 


3- 9 3 


85 5o 


0.229 


o-36 


89 4o 


2-241 


4-54 


86 


0.248 


0-39 


89 5o 


2.549 


5-26 


86 10 


0.269 


0-43 


90 


— 2.909 


+ 6.12 


86 20 


— 0*292 


+ 0.47 









The column marked T is to be multiplied by (t — 50°) ; and the column marked 
B is to be multiplied by (6 — 3o in -oo). The results are to be applied to the ap- 
proximate refraction obtained by the preceding tables. 



432 



SPHERICAL ASTRONOMY. 



TABLE IV. 

For the Equation of Equal Altitudes of the Sun. 



I nterval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 




b. m. 






h. m. 






h. m. 








2 O 


7.7297 


7.7146 


3 


7-7359 


7'70l5 


4 


7.7447 


7.6823 




2 


.7298 


.7143 


2 


•7362 


•7010 


2 


•745i 


•68i5 




4 


• 73oo 


.7139 


4 


• 7 364 


•70o5 


4 


•7454 


.6807 | 


6 


• 7302 


• 7 i36 


6 


•736 7 


• 6999 


6 


• 7 458 


.6800 j 


8 


•73o4 


• 71 32 


8 


. 7 36 9 


•6993 


8 


• 7 46i 


.6792 


IO 


• 7 3o5 


• 7128 


10 


• 7 3 7 2 


•6988 


10 


•7464 


►6 7 84 




12 


.7307 


•7125 


12 


•7374 


.6982 


12 


• 7468 


.6776 




i4 
i 


.7309 


• 7121 


i4 


.7377 


.6976 


i4 


.7472 


.6768 




16 


• 73n 


.7117 


16 


. 7 38o 


•6970 


16 


•7475 


• 6 7 5 9 




18 


• 73i3 


• 7ii3 


18 


• 7 383 


.6964 


18 


•7479 


•6 7 5i 




20 


• 7 3i5 


.7109 


20 


• 7 386 


• 6 9 58 


20 


•7482 


•6 7 43x 




22 


.7317 


• 7io5 


22 


• 7 388 


• 6952 


22 


■ 7 486 
.7490 


_^f34 




24 


.7319 


•7101 


24 


.7391 


.6946 


24 


.6726 




26 


• 7321 


.7097 


26 


. 7 3 9 4 


.6940 


26 


•7494 


.6717 




! * 


• 7 3 2 3 


•7092 


28 


.7397 


.6934 


28 


•7497 


.6708 




3o 


. 7 3 2 5 


•7088 


3o 


• 74oo 


.6927 


3o 


«75oi 


• 6700 




32 


• 7 32 7 


•7083 


32 


• 74o3 


• 6921 


32 


• 75o5 


• 6691 




U 


.7329 


.7079 


34 


.7406 


• 6914 


• 34 


•7509 


.6682 




36 


• 7 33i 


•7075 


36 


.7409 


• 6908 


36 


• 7 5i3 


• 66 7 3 




38 


• 7 333 


.7070 


38 


.7412 


.6901 


38 


.7517 


• 6663 




4o 


• 7 336 


• 7g65 


4o 


•74i5 


.6894 


4o 


• 7521 


• 6654 




42 


• 7 338 


•7061 


42 


•74i8 


.6888 


42 


• 7525 


• 6645 




44 


• 7340 


•7o56 


44 


.7421 


.6881 


44 


.7529 


• 6635 




46 


.7342 


• 7o5i 


46 


•7424 


.6874 


46 


• 7 533 


.6626 




48 


•7345 


• 7046 


48 


.7428 


.6867 


48 


•7537 


.6616 




5o 


.7347 


.7041 


5o 


•743 1 


• 685 9 


5o 


•754i 


.6606 




52 


•7349 


. 7o36 


52 


•7434 


.6852 


52 


• 7 545 


.6597 




54 


• 7 352 


• 7o3i 


54 


.7437 


• 6845 


54 


•7549 


• 658 7 




56 


•7354 


.7026 


56 


•744i 


• 6838 


56 


• 7 553 


•65 7 7 




58 


7.7357 


7.7021 


58 


7.7444 


7 .683o 


58 
1 


7.7557 


7-6567 





TABLES. 



433 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


t 

Log. B. 


h. m. 






h. m. 






h. m. 






5 o 


7-7562 


7.6556 


6 O 


7.7703 


7.6198 


7 


7.7873 


7-5717 


2 


• 7 566 


-6546 


2 


.7708 


•6l84 


2 


.7879 


• 5699 


4 


•7570 


• 6536 


4 


. 77 i3 


.6170 


4 


- 7 885 


-568o 


6 


• 7 5 7 5 


-6525 


6 


.7719 


• 6i56 


6 


.7891 


• 566i 


8 


•7579 


-65i4 


8 


.7724 


• 6142 


8 


.7898 


-564i 


IO 


• 7 583 


.65o4 


10 


.7729 


• 6127 


10 


.7904 


.5622 


12 


• 7 588 


.6493 


12 


• 77 35 


♦ 6n3 


12 


.7910 


• 56o2 


U 


•7592 


.6482 


H 


•774o 


.6098 


id 


.7916 


-5582 


16 


•7597 


•6471 


16 


• 7745 


-6o83 


16 


.7923 


.5562 


18 


•7601 


• 646o 


18 


• 775l 


.6068 


18 


.7929 


• 5542 


20 


• 7606 


-6448 


20 


•7756 


• 6o53 


20 


.7936 


• 5522 


22 


.7610 


.643 7 


22 


.7762 


-6o38 


22 


.7942 


• 55oi 


24 


. . 7 6i5 


-6425 


2 f 


.7767 


• 6023 


24 


.7949 


.5480 


26 


• 7620 


.64i4 


26 


.7773 


•6007 


26 


- 79 55 


-545 9 


28 


• 7624 


• 6402 


28 


•7779 


.5991 


28 


.7962 


•543 7 


3o 


• 7629 


.6390 


3o 


-7784 


•5975 


3o 


.7969 


-54i6 


32 


• 7 634 


• 63 7 8 


32 


.7790 


-5 9 5 9 


32 


.7975 


-53 9 4 


34 


•7638 


-6366 


34 


.7796 


-5 9 43 


34 


.7982 


-53 7 2 


36 


•7643 


-6354 


36 


.7801 


•5927 


36 


.7989 


• 535o 


38 


• 7 648 


-6342 


38 


.7807 


•5910 


38 


.7995 


.5327 


4o 


• 7 653 


• 6329 


4o 


• 7 8i3 


-58 9 4 


4o 


• 8002 


• 53o4 


4a 


•7^58 


• 63.17 


42 


.7819 


-58 77 


42 


• 8009 


-52«I 


44 


•7663 


>63o4 


44 


.7825 


-586o 


44 


-8016 


.5258 


46 


.7668 


• 6291 


46 


. 7 83 1 


-5843 


46 


•8o23 


-5234 


48 


.7673 


.6278 


48 


. 7 836 


• 5825 


48 


• 8o3o 


•5211 


5o 


.7678 


-6265 


5o 


.7842 


• 58o8 


5o 


.8037 


• 5i86 


52 


• 7 683 


-6252 


52 


.7848 


.5790 


52 


-8o44 


• 5i62 


54 


.7688 


'6239 i 


54 


•7854 


•5772 


54 


-8o5i 


• 5i3 7 


56 


.7693 


-6225 


56 


.7860 


-5 7 54 


56 


-8o58 


•5lI2 


58 

1 


7-7698 


7-6212 


58 1 
1 


7 • 7867 


7-5 7 36 


58 


7-8o65 


7.5087 



28 



434 



SPHERICAL ASTRONOMY. 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


h. m 






h. m. 






h. m. 




i 


8 o 


7^8072 


7.5062 


9 


7-83o2 


7-4i3i 


IO 


7-8567 


7.2697 


2 


.8079 


• 5o36 


2 


•83u 


• 4093 


2 


-85 7 6 


• 2635 


4 


.8086. 


• 5oio 


4 


• 83i9 


• 4o55 


4 


• 8586 


.2572 


6 


.8094 


.4983 


6 


•83 2 8 


.4016 


6 


• 85 9 5 


• 2507 


8 


.8101 


•4957 


8 


• 8336 


.3977 


8 


• 86o5 


• 2442 


IO 


•8108 


•493o 


10 


.8344 


.8937 


10 


.8614 


•23 7 4 


12 


• 8ll6 


•4902 


12 


• 8353 


.38 9 6 


12 


.8624 


• 23o6 


i4 


• 8l23 


•48 7 4 


i4 


• 836i 


• 3855 


i-4 


• 8634 


•2236 


16 


•8i3o 


• 4846 


16 


.8370 


• 38i3 


16 


• 8643 


• 2164 


18 


• 8i38 


.4818 


18 


• 83 7 8 


.3771 


18 


.8653 


• 2091 "' 


20 


• 8i45 


.4789 


20 


• 8387 


.3728 


20 


• 8663 


• 2C <6 


22 


•8i53 


.4760 


22 


.8396 


.3684 


22 


.8673 


• i£4o 


24 


.8160 


•473i 


24 


• 84o4 


.3639 


24 


• 8683 


••1861 


26 


.8168 


• 4701 


26 


• 84i3 


• 35 9 4 


26 


.8693 


.1781 


28 


.8176 


.4671 


28 


.8422 


• 3548 


28 


• 8703 


.1699 1 


3o 


.8i83 


•464o 


3o 


• 843o 


• 35oi 


3o 


• 8 7 i3 


• i6i5 


32 


.8191 


•4609 


32 


.8439 


.3454 


32 


•8723 


• 1529 


34 


.8199 


.4578 


34 


.8448 


.3406 


34 


• 8 7 33 


• i44o 


36 


.8206 


•4546 


36 


• 845 7 


• 335 7 


36 


• 8 7 43 


•i34 9 


38 


.8214 


•45i4 


38 


•8466 


.3307 


38 


• 8 7 53 


• 1256 


4o 


'8222 


.4482 


4o 


• 84 7 5 


.3256 


4o 


• 8 7 63 


• 1 160 


42 


• 823o 


.4449 


42 


.8484 


• 32o5 


42 


.8773 


.1061 


44 


.8238 


•44i5 


44 


.8493 


• 3i52 


44 


.8784 


• 0960 


46 


8246 


• 438 1 


46 


• 85o2 


.3099 


46 


•8794 


• o855 


48 


.8254 


•4347 


48 


• 85ii 


• 3o45 


48 


.8804 


• o 7 48 


5o 


.8262 


• 43i2 


5o 


• 852o 


.2989 


5o 


• 88i5 


• 0637 


52 


.8270 


.4277 


52 


• 853o 


.2933 


52 


• 8825 


•o522 


54 


.8278 


.4241 


54 


• 853 9 


.2876 


54 


• 8836 


• o4o4 


56 


.8286 


.4205 


56 


.8548 


.2817 


56 


• 8846 


.0282 


58 


7.8294 


7.4168 


58 


7-8558 


7.2758 


58 


7-8857 


7»oi56 



TABLES. 



435 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



Interval 


1 
Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


b. tn. 






h. m. 






h. m. 






II o 


7-8868 


7.0025 


12 O 


7.9208 


B = o 


r3 « 


7.9593 


— 7-0750 


2 


.8878 


6.9889 


2 


.9220 


-5.5549 


2 


.9607 


•0905 


4 


.8889 


•9748 


4 


.9232 


5-8641 


4 


.9620 


•10G6 


6 


• 8900 


• 9602 


6 


.92.45 


6-o4i4 


6 


.9634 


•I203 


8 


• 8911 


•9449 


8 


.9257 


• i6 7 5 


8 


.9648 


• i345 


lO 


• 8922 


• 9290 


10 


• 9269 


.2657 


10 


.9662 


• i484 


12 


.8932 


• 9125 


12 


.9281 


• 346i 


12 


.9676 


• 1619 


i4 


.8943 


• 8 9 53 


i4 


.9294 


•4i42 


i4 


•9690 


• 1751 


16 


• 8 9 54 


.8770 


16 


.9306 


.4734 


16 


.9704 


. 1880 


18 


.8965 


• 858o 


18 


•9 3l 9 


• 5258 


18 


.9718 


• 2006 


* 20 


.8977 


.83 79 


20 


.9331 


.5728 


20 


• 97 32 


• 2129 


! 22 


.8988 


.8168 


22 


.9344 


• 6i54 


22 


.9746 


.2249 


24 


.8999 


•7945 


24 


. 9 35 7 


• 6545 


24 


.9761 


. 2 36 7 


26 


.9010 


• 7709 


26 


.9369 


• 6905 


26 


•9775 


.2482 


28 


.9021 


•745 7 


28 


.9382 


.72J9 


28 


.9789 


2 5 9 5 


3o 


• 9033 


.7189 


3o 


• 93 9 5 


• 7 55i 


3o 


.9804 


.2706 


32 


.9044 


.6901 


32 


.9408 


.7843 


32 


.9818 


.2815 


34 


.9055 


.6591 


34 


.9421 


.8119 


34 


.9833 


.2922 


36 


.9067 


• 6255 


36 


.9433 


• 838o 


36 


.9848 


.3026 


38 


.9078 


.5889 


38 


• 9446 


.8627 


38 


.9862 


.3129! 

! 
1 


4o 


•9090 


.5487 


4o 


.9460 


.8863 


4o 


.9877 


1 
.323i| 


42 


.9102 


• 5o4i 


4* 


•9473 


.9087 


42 


.9892 


.333c 1 


44 


.9113 


•454i 


44 


.9486 


.9302 


44 


.9907 


.3428 1 


46 


•9125 


• 3 97 3 


46 


.9499 


•9 5 °7 


46 


.9922 


• 3524 


•48 


. 9 i3 7 


.33i6 


48 


• 9 5l2 


.9705 


48 


.9937 


.3619 


5o 


.9148 


• 2536 


5o 


.9526 


6.9895 


5o 


• 99 52 


.3712' 


52 


• 9160 


.1579 


5a 


. 9 53 9 


7.0078 


52 


.9967 


•38of | 


54 


.9172 


6-o34i 


54 


• 9552 


• 0254 


54 


.9982 


• 38 9 4 


56 


.9184 


5-85 9 3 


56 


• 9 566 


•o425 


56 


7.9998 


.3984 


58 


7.9196 


5.5594 


58 


7.9580 


— 7-0590 


58 


8-ooi3 


- 7-4o7i 



436 



SPHERICAL ASTRONOMY. 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


h. m. 

i4 o 


8- 0028 


- 7*4*58 


h. m, 
l5 


8-o52i 


- 7-635o 


h. m. 
16 O 


8.1082 


— 7.8072 


2 


• oo44 


•4244 


2 


• o53 9 


•64i3 


2 


•1102 


• 8i25 


4 


• 0059 


.4328 


4 


• o556 


•64 7 5 


4 


•1122 


.8177 


6 


• 0075 


.4412 


6 


•0574 


•653 7 


6 


•Il43 


•8229 


8 


• 0090 


•4494 


8 


• 0592 


•65 99 


8 


•n63 


• 82bi 


IO 


.0106 


•45 7 5 


10 


• 0610 


.6660 


10 


• ii83 


• 8333 j 


12 


•0122 


• 4655 


12 


.0628 


• 6721 


12 


• 1204 


.8385 


i,4 


• oi38 


•4735 


i4 


.0646 


.6781 


i4 


• 1224 


.8436 


16 


• oi54 


• 48i3 


16 


• o664 


.6841 


16 


•1245 


•848 7 


18 


•0170 


.4890 


18 


.0682 


.6900 


18 


• 1266 


.8538 


20 


•0186 


.4967 


20 


• 0700 


.6 9 5 9 


20 


• 1287 


.8589 


22 


•0202 


.5o43 


22 


.0718 


.7018 


22 


-i3o8 


• 864o 


24 


• 0218 


• 5u8 


24 


.o 7 3 7 


.7077 


24 


'1329 


.8690 


26 


• 0234 


• 5192 


26 


.0755 


• 7 i35 


26 


•i35o 


.8740 


28 


•0250 


• 5265 


28 


.0774 


.7192 


28 


•1371 


.8790 


3o 


.0267 


• 5338 


3o 


.0792 


.7249 


3o 


•1393 


• 884o 


32 


•0283 


• 54io 


32 


.0811 


.7306 


32 


•i4i4 


.8890 


34 


•o3oo 


• 548i 


34 


.o83o 


• 7 363 


34 


•i436 


_-- 8 9 3 9 


36 


•o3i6 


• 555i 


36 


•0849 


.7419 


. 36 


•i^f 


.8989 


38 


•o333 


.5621 


38 


.0868 


•7475 


38 


•1479 


.9038 


4o 


•o35o 


.5690 


4o 


.0887 


• 7 53i 


4o 


• i5oi 


.9087 j 


4a 


• 0367 


.5759 


42 


.0906 


• 7 586 


42 


•i5 2 3 


.91 36 


M 


• o384 


.5827 


44 


.0925 


.7641 


44 


.i545 


• 9 i85 


46 


• o4oo 


• 58 9 4 


46 


.0945 


7696 


46 


• i568 


.9234 


48 


.0417 


.5961 


48 


.0964 


; 7 5i 


48 


1590 


.9282 j 


5o 


• o435 


• 6027 


5o 


.0983 


. 7 8o5 


5o 


• 161 2 


• 933o 1 


52 


• o452 


.6092 


52 


• ioo3 


. 7 85 9 


52 


•1 635 


•9 3 79! 


54 


• 0469 


• 6i58 


54 


•I023 


.7912 


54 


•i658 


.9427 1 


56 


• o486 


.6222 


56 


.1042 


.7966 


56 


• i68o 


.9475 1 


58 


8-o5o4 


- 7.6286 


58 


8-1062 


— 7.8019 


58 

1 


8-1703 


- 7-9^23 1 
j 



TABLES. 



437 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



r" 

Interval L 


og. A. 


Log. B. 


Interval L 


og. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


b. m. 
17 o 8 


•1726 


- 7-9 5 7 r 


h. m. 

18 O 8 


• 2474 


— 8*0969 


h. m. 
19 O 


8.3359 


-8-2354 


2 


.1749 


.9618 


2 


. ?.5oi 


• ioi5 


2 


• 3392 


• 2401 


4 


.1773 


.9666 


4 


.2529 


• 1061 


4 


.3424 


.2448 


6 


1796 


. 97 i3 


6 


.2556 


.1107 


6 


•3457 


.2495 


8 


1819 


• 9761 


8 


2583 


• n53 


8 


.3490 


.2542 


IO 


l843 


.9808 


10 


261 1 


• 1199 


10 


• 3524 


.2589 


12 


1867 


• 9 855 


12 


2639 


.1245 


12 


•355 7 


• 2637 


i4 


[890 


• 9902 


i4 


2667 


.1291 


i4 


• 35 9 l 


.2684 


16 


1914 


• 99 4 9 


16 


2695 


.i336 


16 


• 36 2 5 


• 2732 


18 


1938 


7.9996 


18 


2723 


• i382 


18 


• 365 9 


.2779 


20 


1963 


8-oo43 


20 


2752 


.1428 


20 


•36 9 4 


.2827 


22 


1987 


.0090 


22 


2781 


•i474 


22 


.3728 


.2875 


24 


2011 


• 0137 


24 


2809 


•l520 


24 


• 3 7 63 


.2923 


26 


2o36 


.0184 


26 


2838 


-1 566 


26 


.3798 


.2971 


28 . 


2061 


• 0230 


28 


2868 


.1612 


28 


• 3834 


.3019 


3o . 


2086 


.0277 


3o 


2897 


•1 658 


3o 


.3869 


• 3o68 


32 . 


2111 


• o323 


32 


2926 


•1704 


32 


.3905 


.3ii6 


34 • 


2i36 


• 0370 


34 


2966 


•1750 


34 


.3941 


• 3i65 


36 . 


2161 


• 0416 


36 


2986 


•1797 


36 


.3978 


.3214 


38 . 


2186 


• 0462 


38 


3oj6 


•1842 


38 


• 4oi5 


• 3263 


4o . 


2212 


• o5o8 


4o 


3o46 


•1889 


4o 


• 4o52 


• 33i2 


42 . 


2237 


• o555 


42 


3077 


• i 9 35 


42 


.4089 


• 336i 


44 • 


2263 


• 0601 


44 


3107 


•1 981 


44 


.4126 


• 34io 


46 . 


2289 


.0647 


46 


3i38 


.2028 


46 


• 4i64 


• 346o 


48 . 


23i5 


.0693 


48 ' 


3169 


•2074 


48 


.4202 


.35io 


5c • 


234c 


.0739 


5o . 


3200 


.2121 


5o 


.4241 


•356o! 


52 


236 7 


• o 7 85 


52 . 


3232 


♦ 2167 


52 


.4279 


1 
• 36io 


54 • 


2394 


• o83i 


54 • 


3 2 63 


•22l4 


54 


• 43i8 


• 366o 


56 • 


2420 


.0877 


56 • 


32 9 5 


.226l 


56 


•435 7 


•3711^ 


58 8. 


2447 


— 8.0923 


58 8- 


3327 


— 8.2307 


58 


8.4397 


- 8.3761 



438 



SPHERICAL ASTRONOMY. 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log.B. 


h, m. 
20 O 


8.4437 


- 8-3812 


h. m. 
21 


8-58io 


- 8-5466 


h. m. 
22 O 


8.7711 


-8. 7 56o 


2 


•4477 


.3863 


2 


• 5863 


•5527 


2 


.7789 


•7643 


4 


•45i8 


• 3 9 i5 


4 


.5917 


• 5588 


4 


.7868 


•7727 


6 


.4559 


.3966 


6 


•5971 


• 565o 


6 


•7948 


• 7 8i3 


8 


• 4600 


.4018 


8 


.6025 


.5712 


8 


.8o3o 


.7899 


10 


•464i 


.4070 


10 


.6081 


•5 77 5 


10 


• 8u3 


.7987 


12 


•4683 


•4l22 


12 


.61 36 


.5838 


12 


.8198 


.8076 


M 


.4726 


•4175 


i4 


• 6193 


.5902 


i4 


.8284 


.8167 


16 


.4768 


.4227 


16 


• 625o 


.5966 


16 


• 83 7 2 


.8259 


18 


.4811 


.428o 


18 


• 63o8 


.6o3i 


18 


• 846i 


.8353 


20 


•4854 


.4334 


20 


.6366 


.6096 


20 


• 8553 


.8448 


22 


.4898 


• 438 7 


22 


.6426 


.6162 


22 


•8645 


.8545 


24 


.4942 


•4441 


24 


.6486 


.6229 


24 


•8 7 4o 


.8644 


26 


•4987 


.4495 


26 


• 6546 


.6296 


26 


•883 7 


.8745 


28 


• 5o32 


.4549 


28 


.6608 


.6364 


28 


•8935 


.8847 


3o 


.5077 


•46o4 


3o 


• 6670 


.6433 


3o 


• ox>36 


.8952 


32 


• 5i23 


•465 9 


32 


• 6 7 33 


• 65o2 


32 


•9 l3 9 


.9058 


34 


•5169 


.47i4 


34 


.6796 


.6572 


34 


.9244 


.9167 


36 


• 52i5 


.4770 


36 


.6861 


.6643 


36 


• 9 35i 


.9278 


38 


• 5262 


.4826 


38 


.6927 


.6715 


38 


• 9 46i 


. 9 3 9 i 


4o 


• 53io 


.4882 


4o 


• 6993 


.6788 


4o 


•9574 


•9 5 °7 


42 


•535 7 


•4 9 3 9 


42 


• 7060 


.6860 


42 


.9689 


.9626 


44 


.54o6 


.4996 


44 


.7128 


.6934 


44 


.9807 


•9747 


46 


• 5455 


-5r S3 


46 


.7197 


.7009 


46 


8.9928 


.9871 


48 


• 55o4 


• r ..n 


48 


.7268 


.7085 


48 


9>oo52 


8.9999 


5o 


• 5554 


• 5i6 9 


. 5o 


.7339 


.7162 


5o 


• 0180 


9.0129 


52 


• 56o4 


.5228 


52 


•74ii 


•7239 


52 


•o3u 


.0263 


54 


• 5655 


.5287 


54 


•7484 


• 7 3i8 


54 


•o446 


.0401 


56 


•5706 


• 5346 


56 


• 7 558 


.7398 


56 


•o585 


•o543 


58 


8.5 7 58 


-8.54o6 


58 


8. 7 634 


- 8.7478 


58 


9.0729 


— 9-0689 



TABLES. 



439 



Table IV. — (Continued.) 
For the Equation of Equal Altitudes of the Sun. 



Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


Interval 


Log. A. 


Log. B. 


b. m. 
23 O 


9-0877 


— 9.0839 


h. m. 
23 20 


9.2693 


- 9.2677 


h. m. 
23 4o 


9.5761 


— 9-5757 


2 


•1029 


.0995 


22 


.2922 


•2907 


42 


• 6224 


• 6221 


4 


.1187 


-n55 


24 


• 3l62 


• 3i49 


44 


•6742 


-6739 


6 


-i35i 


•l32I 


26 


• 34i6 


• -34o4 


46 


.7328 


.7326 


8 


• l520 


.1492 


28 


• 3685 


• 36 7 4 


48 


-8oo3 


-8001 


IO 


• 1696 


• 1670 


3o 


.3971 


• 3962 


5o 


.8801 


.8800 


12 


.1879 


-i855 


32 


• 4276 


.4268 


52 


9.9776 


9.9775 


i4 


.2069 


. -2047 


34 


.4604 


•4597 


54 


o-io3i 


o-io3i 


16 


.2268 


- 2248 


36 


•4957 


.4952 


56 


0-2798 


0-2798 


18 


9*2476 


— 9-2456 j 


38 


9.5341 


-9-5336 


58 


o-58i4 


-o-58i4 



440 



SPHERICAL ASTRONOMY. 



TABLE V. 

For the Reduction to the Meridian : showing the value of 
2 sin 2 i P 

A = ; V— . 

sin 1" 



Sec. 


0™ 


l m 


2 m 


3 m 


4m 


5 m 


6™ 


V Bi 


o 


o-o 


2-0 


7-8 


17.7 


3i-4 


49-1 


70.7 


96-2 


i i 


O-O 


2-0 


8-o 


17.9 


3i-7 


49.4 


71. 1 


96.7 


2 


o-o 


2-1 


8-i 


18. 1 


3i- 9 


49.7 


7 i-5 


97.1 


3 


o-o 


2-2 


8-2 


i8-3 


32-2 


5o.i 


71-9 


97-6 


4 


O'O 


2-2 


8-4 


18.5 


32-5 


5o-4 


72.3 


98.0 


5 


o-o 


2-3 


8-5 


18.7 


32-7 


5o-7 


72-7 


98.5 


6 


o-o 


2-4 


8-7 


18-9 


33- 


5i-i 


7 3-i 


99.0 


7 


o-o 


2-4 


8-8 


19. 1 


33-3 


5i-4 


7 3-5 


99-4 


8 


o-o 


2-5 


8-9 


19.3 


33-5 


51-7 


73.9 


99-9 


9 


O'O 


2-6 


9.1 


19.5 


33-8 


52-1 


74-3 


100.4 


IO 


O'l 


2.7 


9.2 


19.7 


34-1 


52 4 


74-7 


ioo-8 


ii 


O'l 


2-7 


9.4 


19.9 


34-4 


52.7 


7 5.i 


101 «3 


12 


O-I 


2-8 


9-5 


20 • 1 


34-6 


53-1 


75.5 


ioi-8 


i3 


O-I 


2.9 


9.6 


20.3 


34-9 


53-4 


75.9 


I02'3 


i4 


O-I 


3.0 


9.8 


20 «5 


35.2 


53-8 


76.3 


102.7 


i5 


O-I 


3.1 


9.9 


20 .7 


35.5 


54-i 


76.7 


103-2 


16 


O-I 


3.1 


IO-I 


20.9 


35-7 


54.5 


77.1 


103.7 


17 


0-2 


3-2 


IO-2 


21 .2 


36- 


54-8 


77-5 


104.2 


18 


0-2 


3.3 


IO-4 


21 .4 


36-3 


55-1 


77-9 


io4'6 


*9 


0«2 


3.4 


io-5 


21 -6 


36-6 


55.5 


78.3 


io5«i 


20 


0-2 


3.5 


10.7 


21-8 


36-9 


55.8 


78.8 


io5«6 


21 


0-2 


3-6 


io- 8 


22-0 


37.2 


56-2 


79.2 


106. 1 


22 


0-3 


3-7 


II -o 


22-3 


37.4 


56-5 


79.6 


io6>6 


23 


0-3 


3-8 


II -2 


22«5 


37.7 


56-9 


8o-o 


107.0 


24 


0-3 


3-8 


11. 3 


22.7 


38-o 


5 7 -3 


8o-4 


107.5 


25 


0-3 


3.9 


11. 5 


22>9 


38.3 


5 7 .6 


80.8 


io8-o 


26 


0.4 


4-0 


11. 6 


23-1 


38-6 


58-o 


8l3 


108.5 


27 


0-4 


4-i 


11. 8 


23.4 


38-9 


58-3 


81.7 


109.0 


28 


C4 


4.2 


11. 9 


23-6 


39-2 


58. 7 


82.1 


109.5 


29 


0-5 


4-3 


I2-I 


23-8 


39.5 


59-0 


82.5 


IIO-O 

J 



TABLES. 



441 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 

9, sin 2 I P 
A 



2 sin 2 i P 



sin 1 



Sec. 


ra 


l m 


2 m 


,-! 


4 m 


5 m 


6 m 


ftm 


3o 


o-5 


4.4 


12.3 


24*0 


39.8 


59.4 


83- 


// 
IIO-4 


3i 


o-5 


4-5 


12.4 


24.3 


40 • I 


5 9 .8 


83-4 


iiO'9 


32 


o.6 


4.6 


12.6 


24-5 


4o-3 


6o- 1 


83-8 


111. 4 


33 


o-6 


4-7 


12.8 


24-7 


4o-6 


6c5 


84-2 


in. 9 


34 


o-6 


4-8 


12.9 


25«0 


40-9 


6o-8 


84-7 


112. 4 i 

j 


35 


6.7 


4.9 


i3-i 


25-2 


4l «2 


6l»2 


85-i 


112. 9 j 


36 


0.7 


5.o 


i3-3 


25-4 


41.5 


61.6 


85-5 


n3.4 


3? 


0.7 


5-1 


i3-4 


25-7 


4i-8 


61-9 


86- 


113.9 


38 


0.8 


5.2 


i3-6 


25.9 


42-1 


62.3 


86-4 


n4-4 


! 3 9 


o-8 


5-3 


i3-8 


26-2 


42.5 


62.7 


86-8 


ii4«9 


4o 


0.9 


5-4 


i4-o 


26.4 


42.8 


63- 


87.3 


ii5-4 


4i 


0.9 


5-6 


14. 1 


26-6 


43.1 


63-4 


87.7 


115.9 


i 42 


I-O 


5-7 


i4-3 


26-9 


43.4 


63-8 


88.1 


116. 4 


43 


1 -o 


5-8 


i4-5 


27.1 


43.7 


64-2 


88-6 


116.9 


44 


i«i 


5.9 


i4-7 


27.4 


44-o 


64-5 


89-0 


H7-4 


45 


1 «i 


6-o 


i4-8 


27.6 


44.3 


64-9 


O9.5 


117. 9 


46 


1-2 


6-i 


i5-o 


27.9 


44-6 


65-3 


89.9 


118-4 


47 


1*2 


6-2 


l5-2 


28.1 


44-9 


65. 7 


90-3 


118. 9 


48 


1-3 


6-4 


i5-4 


28.3 


45-2 


66- 


90-8 


119. 5 


49 


i-3 


6-5 


i5-6 


28-6 


45-5 


66-4 


91 -2 


I20«0 


5o 


1.4 


6-6 


i5-8 


28-8 


45-9 


66-8 


9I.7 


120.5 


5i 


i-4 


6.7 


i5. 9 


29-1 


46-2 


67-2 


92- I 


121 -O 


52 


1.5 


6-8 


16. 1 


29.4 


46-5 


67.6 


92-6 


121 -5 


53 


1.5 


7.0 


i6.3 


29.6 


46-8 


68-0 


93-0 


122-0 


54 


1.6 


7.1 


i6-5 


29 9 


4i-i 


68-3 


93.5 


122-5 


55 


1.6 


7.2 


16.7 


3o-i 


4i 5 


68.7 


93.9 


I23« I 


56 


1.7 


7-3 


1' .9 


3o-4 


47-8 


69-1 


94-4 


123-6 


57 


1.8 


7-5 


17. 1 


3o-6 


48.1 


69.5 


94-8 


I 24. I 


58 


1.8 


7-6 


17.3 


3o«9 


48-4 


69.9 


95.3 


124-6 


59 


1.9 


7-7 


i 7 -5 


3i-i 


48-8 


'70.3 


95.7 


I25-I 


















J 



442 



SPHERICAL ASTRONOMY. 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 
2 sin 9 1 P 
A ~ "linT 77- - 



See. 


gm 


9 ™ 


10 m 


ll m 


12 m 


13 m 


14 m 


o 


125*7 


159-0 


196.3 


23 7 -5 


282.7 


33i.8 


384-7 


i 


126-2 


159.6 


197.0 


238-3 


283-5 


332-6 


385-6 


2 


126-7 


1 60 -2 


197.6 


239-0 


284-2 


333.4 


386-6 


3 


127-2 


l6o-8 


198-3 


239.7 


285.0 


334-3 


38 7 -5 j 


4 


127-8 


l6l.4 


198-9 


24o-4 


285-8 


335-2 


388-4 ! 


5 


128-3 


l62-0 


199.6 


24l '2 


286-6 


336-o 


389.3 ! 


1 6 


128-8 


l62-6 


200 -3 


241 -9 


287-4 


336-9 


390.2 


7 


129-3 


[63-2 


200-9 


242-6 


288-2 


337.7 


391. 1 i 


8 


129.9 


1 63- 8 


201 «6 


243.3 


289.0 


338-6 


392-1 i 


9 


i3o-4 


164.4 


202 -2 


244-1 


289-8 


339.4 


393.0 


IO 


i3i -o 


i65-o 


202-9 


244-8 


290.6 


34o -3 


3 9 3. 9 


ii 


i3i-5 


i65-6 


2o3«6 


245-5 


291-4 


34i -2 


3 9 4-8 


12 


l32-0 


166-2 


204-2 


246-3 


292.2 


342-0 


3 9 5-8 


i3 


i3 2 -6 


166.8 


204-9 


247 -o 


293.0 


342.9 


396.7 


M 


i33-i 


167.4 


2o5-6 


247-7 


2 9 3- 8 


343-7 


397.6 


i5 


i.33,6 


168-0 


206 -3 


2480 


294-6 


344-6 


3 9 8- 6 


16 


134.2 


x68-6 


206-9 


249-2 


295-4 


345-5 


399.5 


17 


i34-7 


169-2 


207 • 6 


249-9 


296-2 


346-4 


4oo -5 


18 


i35-3 


169-8 


208-3 


25o-7 


297-0 


347-2 


4oi-4 


J 9 


i35-8 


170.4 


208 .9 


2 5i-4 


297-8 


348-1 


4o2-3 


20 


i36-3 


171 «o 


209-6 


2D2-2 


298-6 


349-0 


4o3-3 


21 


i36«9 


171 «6 


2IO-3 


253-o 


299-4 


349-8 


4o4-2 


22 


1 i3 7 .4 


172.2 


211 -O 


253-6 


3oo-2 


35o-7 


4o5-i 


23 


i38-o 


172.9 


211 -7 


254-4 


3oi -o 


35i-6 


4o6-o 


24 


138-5 


173.5 


212-3 


255-1 


3oi-8 


352. '5 


407-0 


25 


139-1 


174 1 


2l3«0 


255-9 


3o2-6 


353.3 


408.0 


26 


1 39 -6 


i74 ' 


2i3«7 


256-6 


3o3-5 


354-2 


408.9 


27 


i4o-2 


175.3 


214.4 


25 7 -4 


3o4-3 


355-1 


409.9 


28 


i4o-7 


175.9 


2l5'I 


258-1 


3o5-i 


356- 


4io-8 


29 


i4i-3 


176-6 


2i5-8 


258.9 


3o5- 9 


356-9 


4n-7 



V 



TABLES. 



US 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 
2 sin 2 1 P 



A = 



sin 1' 



Sec. 
3c 


gm 


9 m 


10 m 


ll m 


12 m 


13 m 


14 m 


i4i-8 


177.2 


216.4 


359-6 


3o6-7 


357-7 


412.7 


3r 


142.4 


177-8 


217-1 


260.4 


307.5 


358 


6 


4i3.6 


32 


i43«o 


178.4 


217-8 


261 .1 


3o8-4 


35 9 


5 


4i4«6 


33 


i43.5 


179.0 


218.5 


261 .9 


309-2 


36o 


4 


4i5-5 


34 


i44-i 


179.7 


219-2 


262-6 


3io-o 


36i 


3 


4i6-5 


35 


i44-6 


i8c3 


219.9 


263-4 


3ro-8 


362 


2 


4i7-5 


36 


145.2 


180.9 


220-6 


264.1 


3il6 


363 


r 


418.4 


37 


i45.8 


181. 6 


221 «3 


264.9 


3i2.5 


364 





419.4 


38 


i46-3 


182.2 


222-0 


265.7 


3i3.3 


364 


8 


420.3 


3 9 


146.9 


182.8 


222-7 


266.4 


3i4-i 


365 


7 


421.3 


4o 


i47-5 


i83.5 


223-4 


267.2 


3i5-o 


366 


6 


422-2 


4i 


i48-o 


184. 1 


224*1 


267.9 


3i5-8 


36 7 


5 


423.2 


42 


1 48-6 


i84-7 


224-8 


268.7 


3i6-6 


368 


4 


424-2 


43 


149.2 


i85-4 


225*5 


269.5 


3i 7 -4 


36 9 


3 


425-i 


44 


149-7 


186-0 


226.2 


270.3 


3i8-3 


370 


2 


426.1 


45 


i5o-3 


186-6 


226.9 


271 -o 


319. 1 


3 7 i 


1 


427-0 


46 


150.9 


187.3 


227-6 


271.8 


3i 9 . 9 


372 





428.0 


47 


i5r-5 


187.9 


228.3 


272.6 


320. 8 


3 7 2 


9 


429-0 


48 


l52«0 


188.5 


229.O 


273.3 


321-6 


3 7 3 


8 


429-9 


49 


1 52. 6 


189.2 


229.7 


274-1 


322-4 


3 7 4 


7 


43o«9 


5o 


1 53. 2 


189.8 


23o«4 


274.9 


323-3 


3 7 5 


6 


43i-9 


5r 


I53..8 


190.5 


23l .1 


275.6 


324-r 


3 7 6 


5 


432-8 


52 


i54-4 


191 .1 


2 3i-8 


276.4 


325.o 


377 


4 


433-8 


53 


154.9 


191. 8 


232.5 


277.2 


325-8 


3 7 8 


3 


434-8 


54 


i55-5 


192.4 


233-2 


278.0 


326.7 


379 


3 


435.8 


55 


i56,i 


193. 1 


234-o 


278.8 


327-5 


38o 


2 


436. 7 


56 


i56. 7 


193.7 


234-7 


279.5 


328-4 


38 1 


1 


43 7 -7 


57 


157.3 


194.4 


235-4 


280.3 


329.2 


382 





438. 7 


58 


157.8 


195.0 


236-1 


281 . 1 


33o.o 


382 


9 


439.7 


5 9 


i58-4 


195.7 


236-8 


281.9 


33o>9 


383 


8 


44o-6 



444 



SPHERICAL ASTRONOMY. 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 
2 sin 2 1 P 



A = 



sin 1' 



Sec. 


15 m 


16 m 


I7 m 


18 m 


19 m 


20™ 


- 
21 m 


o 


44i-6 


5o2-5 


567-2 


635-9 


708.4 


784.9 


865-3 


i 


442-6 


5o3.5 


568.3 


637.0 


709-7 


786.2 


866-6 


2 


443-6 


5o4-6 


56 9 .4 


638-2 


710-9 


787-5 


868.0 


3 


444-6 


5o5-6 


5 7 o-5 


63 9 -4 


712-1 


788-8 


869.4 


4 


445-6 


5o6-7 


571-6 


64o-6 


7i3.4 


790.1 


870-8 


5 


446-5 


507.7 


5 7 2-8 


64t-7 


7i4-6 


791.4 


872-1 


6 


447-5 


5o8-8 


5 7 3- 9 


642-9 


715.9 


792.7 


873.5 


7 


448-5 


509-8 


575-o 


644-i 


717-1 


794.0 


874.9 


8 


449-5 


510-9 


576-1 


645-3 


718-4 


795.4 


876.3 


9 


45o-5 


5n -9 


577-2. 


646-5 


7 r 9 -6 


796-7 


877-6 


10 


45i-5 


5i3-o 


5 7 8-4 


647-7 


720-9 


798.0 


879-0 


ii 


452-5 


5i4-o 


579-5 


648.9 


722-1 


799.3 


880.4 


12 


453-5 


5i5-i 


58o-6 


65o-o 


723-4 


800.7 


881 -8 


i3 


454-5 


5i6-i 


58i- 7 


65i-2 


724.6 


8o2«o 


883-2 


| i4 


455-5 


5i7-2 


582.9 


652-4 


725.9 


803.3 


884-6 


i5 


456.5 


5i8.3 


584- 


653-6 


727.2 


8o4-6 


886- 


16 


45 7 -5 


5i 9 -3 


585-1 


654-8 


728-4 


8ob- 


887.4 


1 I7 


458-5 


52o-4 


586-2 


656 -o 


729-7 


807.3 


888-8 


1 «« 


45 9 -5 


521.5 


58 7 -4 


657-2 


7 3o-9 


808 -6 


890.2 


! '9 


46o.5 


522-5 


588-5 


658-4 


732-2 


809-9 


891.6 


20 


46i-5 


523-6 


58 9 -6 


65 9 -6 


7 33-5 


8u-3 


893.0 


21 


462-5 


524-6 


590.8 


660.8 


734-7 


812-6 


894 -4 


22 


463-5 


525.7 


591.9 


662-0 


736 -o 


8i3-9 


895.8 


! 23 


464-5 


526-8 


593-0 


663-2 


7 3 7 -3 


8i5-2 


897.2 


24 


465-5 


527.9 


594-2 


664-4 


7 38-5 


816.6 


898-6 


25 


466-5 


528-9 


595-3 


665-6 


7 3 9 .8 


817.9 


900-0 


26 


467.5 


53o«o 


596.5 


666-8 


741 • 1 


819.2 


901*4 


27 


468-5 


53i-i 


597.6 


668- 


742.3 


820.5 


902*8 


28 


469.5 


532-2 


598.7 


669*2 


743-6 


821.9 


904-2 


29 


47o.5 


533.2 


599.9 


670.4 


744.9 


823.2 


905. 6 



TABLES 



445 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 



Sec. 


15 m 


16 m 


I7 ra 


18 m 


19 m 


20 ra 


1 

21 m 


3o 


47i-5 


534.3 


6oi 'O 


671.6 


746-2 


824 6 


907-0 


3i 


472 6 


535-4 


602-2 


672.8 


747.4 


825.9 


908 


4 


32 


473 


6 


536-5 


6o3-3 


6 7 4 • 1 


748-7 


827.3 


909 


8 


33 


474 


6 


53 7 -6 


6o4-5 


675-3 


75o«o 


828-6 


911 


2 


M 


4 7 5 


6 


538-7 


6o5.6 


676.5 


7 5i-3 


829.9 


912 


6 


35 


476 


6 


539.7 


606 -8 


677.7 


752-6 


83i-2 


914 





36 


477 


6 


54o-8 


607-9 


678-9 


7 53- 8 


832.6 


915 


5 


3? 


478 


7 


54i-9 


609. i 


680 • i 


755-1 


833-9 


916 


9 


38 


479 


7 * 


543.o 


6lO-2 


68i-3 • 


756-4 


835-3 


918 


3 


3 9 


48o 


7 


544-1 


6ii- 4 


682.6 


757.7 


836-6 


919 


7 


4o 


48i 


7 


545.2 


612 -5 


683-8 


759-0 


838- 


921 


1 


4i 


482 


8 


546-3 


613.7 


685- 


760-2 


83 9 -3 


922 


5 


42 


483 


8 


547-4 


6i4-8 


686.2 


761.5 


84o- 7 


923 


9 


43 


484 


8 


548-4 


616-0 


687.4 


762.8 


842.0 


925 


3 


44 


485 


8 


549-5 


617-2 


688.7 


764.1 


843-4 


926 


8 


45 


486 


9 


55o.6 


6i8.3 


689.9 


765.4 


844-7 


928 


2 


46 


48 7 


9 


55i-7 


619.5 


691 «i 


766.7 


846.1 


929 


6 


47 


488 


9 


552-8 


620.6 


692.4 


768.0 


847-5 


9 3i 





48 


490 





553.9 


621-8 


693.6 


769.3 


848.9 


932 


4 


49 


491 





555.o 


623-0 


694-8 


770.6 


85o-2 


9 33 


8 


5o 


492 





556-1 


624.1 


696-0 


771.9 


86i-6 


9 35 


2 


5i 


493 


1 


557*2 


625.3 


697.3 


773-1 


852.9 


9 36 


6 


52 


494 


1 


558-3 


626.5 


698.5 


774-5 


854-3 


9 38 


1 


53 


495 


2 


559.4 


627.6 


699.7 


775-8 


855-7 


9 3 9 


5 


54 


496 


2 


56o.5 


628.8 


701.0 


777.1 


857-1 


940 


9 


55 


497 


2 


56i-6 


63o«o 


702 > 2 


778.4 


858-4 


942 


3 


56 


498 


3 


562 7 


63r-2 


7o3-5 


779*7 


859-8 


943 


8 


57 


499 


3 


563-9 


632-3 


7o4 -7 


781-0 


86i- 1 


945 


2 


58 


5oo 


3 


565- 


633-5 


705.? 


782.3 


862-5 


946 


6 


59 


5oi 


4 


566-1 


634-7 


707.: 


7 83- 6 


863-9 


948 


• 1 



r&V ftsO-e* f 



'Sls'V^'L/ L>7' 



446 



SPHERICAL ASTRONOMY. 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 
2 sin 2 i P 



A = 



sin 1" 



Sec. 


22 m 


23 m 


24 m 


25 m 


26 m 


27 ra 


28 ra 




o 


949.6 


1037.8 


1129.9 


1225-9 


r325»9 


1429 


7 


i53 7 .5 




i 


95l 'O 


1039.3 


1 i3i -4 


1 227 -5 


i32 7 -6 


i43i 


4 


i53 9 -3 




2 


952.4 


io4o«8 


n33-o 


1229-2 


1329.3 


i433 


2 


i54it 




3 


953.8 


1042.3 


n34-6 


i23o-8 


i33i «o 


i434 


9 


1542-9 




4 


955.3 


io43.8 


n36-2 


1232-5 


i332- 7 


1 436 


7 


i544-8 


J 


5 


956-7 


io45.3 


n3 7 -8 


t234- 1 


1334-4 


i438 


5 


i546-6 




6 


9 58,2 


1046.8 


1139-3 


1235-7 


i336-i 


1 44o 


3 


i548-4' 




7 


9 5 9 .6 


io48.3 


n4o-9 


1237-3 


i33 7 -8 


1442 


1 


i55o-2 




8 


961 «i 


1 049 - 8 


ri42-5 


1239-0 


i33 9 -5 


i443 


9 


[552.x 




9 


962-5 


io5i -3 


ii44-o 


1240-6 


i34i-2 


i445 


6 


i553. 9 




IO 


963.9 


io52-8 


1145-6 


1242-3 


1342-9 


1 447 


4 


1555-8, 




ii 


965.4 


io54-3 


n47-2 


1243-9 


i344-6 


1449 


2 


i55 7 -6 


: 


12 


966-9 


io55«9 


n48-8 


1245-6 


1346-3 


i45i 


• 


i55 9 -5 


i3 


968.3 


1057.4 


1160.4 


1247-2 


i348-o 


1 45 a 


■ 8 


i56i-3 




i4 


969.8 


io58-9 


II52-0 


1248-9 


1349-7 


i454 


• 5 


i563-2 




i5 


971.2 


io6o-4 


n53-6 


i25o-5 


i35t-4 


i456 


3 


i565-o 




16 


972.7 


1062-0 


n55-2 


1352-2 


i353-2 


i458 


1 


i566-9 




17 


974-1 


io63-5 


n56-8 


1253-8 


i354-9 


i45 9 


9 


1 568 .'7 




18 


975.5 


io65-o 


ii58-3 


1255-5 


i356-6 


1461 


6 


i5 7 o-5 




*9 


977.0 


1066 -5 


1159.9 


1257-1 


i358-3 


1 463 


4 


i5 7 2-4 




20 


978.5 


1068. 1 


ii6l5 


r258-8 


i36o-i 


i465 


2 


i5 7 4-3 




21 


979-9 


1069-6 


ii63-i 


1260.4 


i36i-8 


1 466 


9 


i5 7 6-i 




22 


981.4 


1071 • 1 


ii64-7 


1262- 1 


i363-5 


1 468 


7 


1578.0 




23 


982.9 


1072.6 


ii66-3 


1263-7 


i365-2 


1470 


•5 


1579-8 




24 


984.4 


1074-2 


1167.9 


1265-4 


1367.0 


1472 


3 


i58l 7 




25 


985.8 


1075.7 


1169-5 


1 267 • 


i368- 7 


i474 





i583-5 




26 


987-3 


1077-2 


1171-1 


1268.7 


1370-4 


i475 


9 


i585-3 




27 


988-8 


1078.7 


1172-7 


1270- 5 


1372-1 


i477 


7 


i58-.2 




28 


990-3 


1080 .3 


ii 7 4-3 


1272. 


1373.9 


1479 


5 


1 58 9 -i 




29 


991.8 


1081.8 


1^75-9 


1273.7 


i3 7 5-6 


1481 


3 


1590^9 





TABLES. 



447 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 
2 sin 2 J P 



A = 



sin 1' 



Sec. 


22 m 


23 m 


24 m 


25 m 


26 m 


27 m 


28" 


3o 


99 3.2 


io83.3 


1 177.5 


1275.4 


1377 


4 


i483-r 


1592.7 


3i 


994-7 


1084 -8 


1179.1 


T277.1 


1379 





i484-9 


i5 9 4-6 


32 


996-2 


1086.4 


1 1 80 • 7 


1278.8 


i38o 


8 


1486.7 


1596-5 


33 


997.6 


1087.9 


1182.3 


1280.4 


]382 


5 


1488-5 


i5 9 8-3 


34 


999.1 


1089.5 


1183.9 


1282. 1 


t384 


2 


1490.3 


I 60O -2 


35 


iooo«6 


1091 -o 


n85-5 


I2.83-8 


i385 


9 


1492. 1 


l602 • I 


36 


1002. 1 


1092-6 


1187.1 


1285-5 


i38 7 


7 


1493-9 


i6o4-o 


37 


ioo3«5 


1094-1 


1188.7 


1 287. 1 


[38 9 


4 


1495.7 


i6o5-9 


38 


ioo5«o 


1095.7 


1190*3 


1288-8 


1391 


2 


i497-5 


1607.7 


3 9 


1006.5 


1097.2 


1191.9 


1290.5 


1392 


9 


1499.3 


1609.6 


4o 


roo8«o 


1098.8 


1193.5 


1292.2 


i3 9 4 


7 


i5oi >i 


i6il5 


4i 


1009.4 


1 100.3 


1 1 95 • 1 


1293-8 


1 3 9 6 


4 


i5o2'9 


i6i3-3 


42 


1010.9 


nor .9 


1196.7 


1295.5 


1 3 9 8 


2 


1 5o4 • 7 


1615.2 


43 


1012.4 


no3-4 


1198.3 


1297.2 


1 3 99 


9 


i5o6-5 


1617. 1 


44 


ioi3«9 


no5«o 


1199.9 


1298-9 


i4o[ 


7 


i5o8-4 


16119.0 


45 


ioi5'4 


1 1 06 • 5 


i2or «5 


1 3oo • 5 


i4o3 


4 


l5lO-2 


1620.8 


46 


1016.9 


1 108. 1 


1203« 1 


l302-2 


i4o5 


2 


l5l2«0 


1622.7 


4? 


1018.4 


1109-6 


1204-7 


i3o3>9 


1 406 


9 


i5i3-8 


1624.6 


48 


1019-9 


mi -2 


1206-4 


i3o5-6 


1408 


7 


i5i5-6 


1626.5 


49 


1021 -4 


1112-7 


1208.0 


i3o7-3 


i4io 


4 


i5i 7 -4 


1628.3 


5o 


1022. 8 


iii4-3 


1209-6 


1 309-0 


1412 


2 


i5i9«2 


i63o«2 


5i 


1024.3 


iii5-8 


I2II -2 


i3io-7 


i4i3 


9 


l52I «0 


i63 2 .i 


52 


1025-8 


1117.4 


I2I2«9 


i3i2-4 


i4i5 


7 


l522«9 


i634-o 


53 


1027.3 


1118-9 


I2l4'5 


i3i4-i 


1417 


4 


1524-7 


i635-9 


54 


1028.8 


n 20 -5 


I2l6« I 


i3i5-7 


1 4 19 


2 


i526-5 


1637.7 


55 


io3o»3 


II22-0 


I217.7 


i3i 7 -4 


[420 


9 


i528-3 


i63 9 -6 


56 


to3i-8 


H23-6 


I219.4 


i3i9« 1 


1422 


7 


i53o-2 


i64i-5 


57 


io33-3 


II25-I 


1221 -O 


i32o-8 


1424 


4 


i532-o 


i643.3 


58 


io34-8 


1126.7 


1222-6 


1322-5 


1426 


2 


i533-8 


i645.2 


59 


io36.3 


1128.3 


1224*2 


l324"2 


1427 


9 


1535-6 


i647-i 



{0}^ - c«X>- 4Aesr>- 



448 



/^f^t,*^ 



SPHERICAL ASTRONOMY. 



Table V. — (Continued.) 
For the Reduction to the Meridian : showing the value of 
2 sin 2 J P 



Sec. 


29 m 


30 m 


31 m 


32™ 


33 m 


34 m 


35 m 


o 


i649«c 


1764.6 


1884.0 


2007.4 


2i34«6 


2265.6 


2400 -6 


i 


i65o«9 


1766.6 


1886-0 


2009-4 


2i36.8 


2267-8 


2402.9 


2 


i 652- 8 


1768.5 


1888.0 


2011 -5 


ot38-9 


2270-0 


24o5«2 


3 


i654-7 


1770.5 


1890-0 


2oi3«6 


2l4l '1 


2272.2 


2407 • 5 


4 


1656-6 


1772.4 


1892-1 


2oi5«7 


2143.2 


2274.5 


2409.8 


5 


1658-5 


1774-4 


1894- 1 


2017.8 


2i45.3 


2276.7 


24l2-0 


6 


i66o-4 


1776.3 


1896-1 


20 1 9,,. 9 


2147.5 


2278-9 


24l4'3 


7 


1662.3 


1778.3 


1898. 1 


2022«0 


2149.7 


228l -2 


2416. 6 


8 


1664.2 


1780.3 


1 900 • 2 


2024-I 


ai5i-8 


2283-4 


2418-9 


9 


1666. 1 


1782.3 


1902.2 


2026.2 


2i53»9 


2285-6 


2421 «2 


IO 


1668-0 


r 7 84-2 


1904.3 


2028.3 


2 1 56- 1 


2287.8 


2423.5 


ii 


1669-9 


1786.2 


1 906 • 3 


2o3o«5 


2i58-3 


2290*0 


2425-8 


12 


1671 .9 


1788-2 


1908-4 


2032»5 


2i6o-5 


2292.3 


2428.1 


i3 


1673.8 


1790. 1 


1910.4 


2o34*6 


2162-6 


2294.5 


2430 .4 


i4 


1675-7 


1792. 1 


1912.4 


2o36«7 


2i64-8 


2296-8 


2432.7 


i5 


1677.6 


1794-1 


1914-4 


2o38-8 


2166-9 


2299-0 


2435.0 


16 


1679.5 


1796-1 


1916-5 


2o4o«9 


2 1 69 - 1 


23oi«3 


2437.3 


. n 


1681.4 


1798-1 


1918.5 


2o43 -o 


2I7T «2 


23o3-6 


2439.6 


8 


1683-3 


1 800 • 


1920-6 


2045 • I 


2173-4 


23o5-8 


2441.9 


l 9 


i685.2 


i8o2-o 


1922*6 


2047 ' 2 


2175-6 


23o8-o 


2444-2 


20 


1687.2 


1804.0 


1924-7 


2049 -3 


2177-8 


23lO«2 


2446-5 


21 


1689. 1 


i8o5-9 


1926-7 


ao5 1 • 4 


2179.9 


23l2«4 


2448-8 


22 


1691-0 


1807.9 


1928-8 


2053.5 


2I82.I, 


23i4-7 


245i-i 


23 


1692-9 


1809-9 


1930-8 


2o55-7 


2l84'3 


23i6-9 


2453-4 


24 


1694-8 


1811.9 


1932.9 


2o57-8 


2186-5 


2319-2 


2455.7 


25 


1696-7 


i8i3-9 


1935-0 


2059-9 


2l88-6 


2321 '5 


2458- 


26 


1698-6 


i8r5.8 


1937-0 


2062-0 


2 1 90 • 8 


2323« 7 


2460 -3 


27 


1700.5 


1817.8 


i 9 3 9 -o 


2064 • 1 


2 1 93 - O 


2325.9 


2462-6 


28 


1702.5 


1819.8 


r 9-4 £ • 1 


2066. 2 


2 1 95 « 2 


2.328-2 


2464.9 


a 9 


1704.4 


1821-8 


1943-1 


2o98-3 


2197.3 


233o-4 


2467-2 






<ZZsC-"0 



TABLES. 



449 



Table V. — (Continued.) 
For the Reduction to the Meridian : 
2 sin 2 1 P 



the value of 



A = 



sin 1' 



i Sec 

I 


29 m 


30™ 


31 ra 


32 m 


33 m 


34 m 


35 m 


1 
3o 


1706.3 


1823-8 


1945.2 


" . 
2070-4 


2199-5 


2332.7 


2469-5 


3r 


1708.2 


1825.8 


1947.2 


2072-6 


2201 -7 


2334-9 


2471-8 


32 


1710.2 


1827.8 


1949-3 


2074-7 


2203«9 


2337-2 


2474-2 


33 


1712.1 


1829.8 


r95i.3 


2076.8 


2206 • 1 


2339-4 


2476.5 


34 


1714-0 


i83i-8 


1953.4 


2078.9 


2208-3 


234i-7 


2478.8 


35 


1715-9 


1833-8 


i 9 55-5 


208 I -o 


22IO-5 


2343-9 


2481. I 


36 


1717-9 


1835-8 


1957-6 


2083-2 


2212-7 


2346-2 


2483-5 


3? 


1719-8 


i83 7 -8 


1959-6 


2o85-3 


22l4'9 


2348-5 


2485-8 


38 


1721-7 


1839.8 


1961-7 


2087-4 


22I7-I 


235o-7 


2488-1 


39 


1723.6 


i84i-8 


1963-7 


2089-6 


22I9-3 


2353-o 


2490-4 


4o 


1725-6 


1843-8 


i 9 65-8 


2091 -7 


2221 -5 


2355-2 


2492-8 


4* 


1727.5 


1845-8 


1967.8 


2093-8 


2223-7 


235 7 -5 


2495 • 1 


42 


1729.5 


1847-8 


1969.9 


2095-9 


2225*9 


2359-7 


2497.4 


43 


i 7 3i.5 


1849-8 


1972.0 


2098 • 


2228-1 


236i -9 


2499.7 


44 


1733.4 


i85i-8 


i974-i 


2IOO-2 


2230-3 


2364 • 2 


25o2«I 


45 


1735.3 


1853.8 


1976- 1 


2102-3 


2232-5 


2366-4 


25o4«4 


46 


I 7 3 7 .2 


1855-8 


1978-2 


2io4-5 


2234-7 


2368-7 


25o6«7 


47 


r 7 3 9 .2 


i85 7 -8 


1980.3 


2106-6 


2236-9 


2371 'O 


2509-0 


48 


1741-2 


1859-8 


1982.4 


2108-8 


2239-1 


2 3 7 3-3 


25n-4 


49 


1743. 1 


1861.8 


i 9 84-4 


2110-9 


2241 -3 


2 3 7 5-5 


25i3-7 


5o 


i 7 45. 1 


1863-8 


1986.5 


2Il3-I 


2243.5 


2377-8 


25i6- 1 


5i 


i747-o 


i865-8 


1988-6 


2Il5«2 


2245.7 


238o-i 


2 5i8-4 


52 


1749-0 


1867.8 


1990.7 


2117.4 


2247.9 


2382-4 


•2520-8 


53 


1750.9 


1869.8 


1992-7 


2119-6 


2250-I 


2384 6 


2523-1 


54 


1752-9 


1871.8 


1994.8 


2121 -7 


2252-3 


2386-9 


2525-4 


55 


i 7 54-8 


i8 7 3-8 


1996-9 


2123-8 


2254-5 


2389-2 


2527-7 


56 


i 7 56-8 


1875.9 


1999.0 


2126-0 


2256-7 


2391 -5 


253o- r 


57 


i 7 58- 7 


1877-9 


2001. 


2128-1 


2258-9 


2393.7 


2532-4 


58 


1760-7 


1879.9 


2003«I 


2i3o-3 


2261-1 


2396-0 


2534-8 


5 9 


1762-6 


1882.0 


2oo5»3 


2l32-4 


2263-4 


2398-3 


2537.1 



// 



-7 



450 



rr 

SPHERICAL ASTRONOMY. 



TABLE VI. 
For the second part of the Reduction to the Meridian : showing the value of 

B 



2 sin 4 \ P 



sin 1" 



Minutes 


s 


10 s 


20 s 


30 s 


40 s 


50 s 


5 


O'OI 


O'OI 


O'OI 


0«0I 


O'OI 


o«oi 


6 


0«OI 


0-01 


O'OI 


0'02 


0»02 


0«02 


7 


0«02 


0-02 


o«o3 


o-o3 


0'o3 


o»o4 


8 


o«o4 


o«o4 


o-o5 


o-o5 


o-o5 


o«o6 


9 


o«o6 


0-07 


o-o8 


o-o8 


0.08 


0.09 


IO 


0*09 


0«I0 


0«II 


O'H 


0'12 


o«i3 


ii 


o«i4 


o.i5 


0'i5 


o«i6 


0-17 


0.18 


12 


0*19 


0-20 


0-22 


0.23 


o«24 


0-25 


i3 


0-27 


0.28 


o-3o 


0'3i 


0.33 


0.34 


i4 


o-36 


0-38 


0'39 


o-4i 


0.43 


0.45 


i5 


o-47 


0.49 


0-52 


o.54 


o-56 


0.59 


16 


o«6i 


o-64 


0-67 


0-69 


0.72 


0.75 


r7 


0.78 


o-8i 


o-84 


o-88 


0.91 


0.95 


■« 


0-98 


I -02 


1. 06 


1*09 


li3 


1. 18 


t 9 


I«22 


1-26 


1 -3o 


i-35 


i-4o 


i-44 


20 


1-49 


1.54 


i«6o 


1-65 


1 .70 


1.76 


21 


1.82 


I.87 


i. 9 3 


1.99 


2- 06 


2«I2 


22 


2.19 


2-25 


2.32 


2.39 


2.46 


2.54 


23 


2«6l 


2.69 


2.77 


2-85 


2.93 


3«OI 


24 


3.io 


3.l8 


3.27 


3-36 


3.45 


3-55 


25 


3-64 


3-74 


3-84 


3 - 9 4 


4-o5 


4-i5 


26 


4-26 


4-3 7 


4-48 


4-6o 


4.72 


4-83 


27 


4.96 


5.08 


5- 20 


5-33 


5-46 


5.6o 


28 


5.73 


5.87 


6>oi 


6-i5 


6-3o 


6-44 


29 


6.59 


6- 7 5 


6-90 


7«o6 


7«22 


7-38 


3o 


7.55 


7.72 


7.89 


8-o6 


8-24 


8.42 


3i 


8.61 


8.79 


8.98 


9.17 


9.37 


9.57 


32 


9*77 


9*97 


io- 18 


10-39 


io«6i 


10.82 


33 


11 «o4 


11.27 


11 «5o 


11 «73 


11.96 


12.20 


34 


i2> 44 


12.69 


12.94 


i3«i9 


i3-45 


13.71 


35 


13.97 


14-24 


i4-5i 


14-78 


i5-o6 


i5.35 



?x ^. ^*^> c^Us*^ 



TRIGONOMETRICAL FOEMUI 



10 



1. 


cos # . tan x. 


2. 


cos # 

cot a? 


3. 


Vl — cos* *. 


4, 


1 




Vl -4-cot 8 ^ 


5 


tan a; 




Vl + tan 2 a; 


6. 


2 sin J re . cos ^ #. 


7. 


^/l — cos 2 x 
V 2 


8. 


2 tan \ x 


1 -f tan* \x' 


9. 


2 


cot ^ # + tan -J x ' 


A 


sin (30° + x) — sin (30 



11. 2 sin 2 (45° + \%) — 1. 

12. 1-2 sin 2 (45° — £ «). 

] - tan 2 (45° - 1 g) 
1 H- tan 2 (45° — | *) * 



13. 



ta n (45° + \ x) — tan (45° - \ x) 
tan (45° + { x) + tan (45° — | x)' 

15. sin (60° + x) — sin {60° — tf). 

16. — L_. 



I. Equivalent expressions for sin as. 




\ 



452 



SPHERICAL ASTRONOMY. 



H. Equivalent expressions for cos 



1, 

2. 
8. 

6. 

7. 
8. 



sin x 
tan # 

sin x . cot av 



Vi 


— sin 8 


A 




1 




Vi 


+ tan 8 
cot X 


X 



VI + cot 8 x 
cos 2 \x — sin 8 J a?. 

1 — 2 sin 8 \ x. 

2 cos 8 \ x — 1. 



10. 
11. 
12. 

la. 

14. 
15. 

16. 



A /\ + cos 2 x 


1 2 


1 — tan 8 \ x 


1 H tan 2 J a;' 


cot \ x — tan \ x 


cot J x + tan J a: * 


1 



1 + tan x . tan J # 
2 



tan (45° + J x) + cot (45° + i *) 
2 cos (45° + \ x) cos (45° — Jar). 
cos (60° -f x) + cos (60° — *). 
1 





secant a? 



TRIGONOMETRICAL FORMULA 453 

m. Equivalent expressions for tan x. 

sin x 
cos x' 

*■ 4- 

QOtX 



3. 


COS 8 iC 


4. 


sin x 




Vl — sin 8 x 


5. 


Vl — cos 8 a? 


cos X 


6, 


2 tan J x 


1 - tan" J a:' 


1. 


2 cot -| a; 


cot 2 J a; — 1* 


ft 


2 



cot £ a; — tan A a? 



9. cot x — 2 cot 2 a?. 
1 — cos 2 a; 



10, 



11. 



sin 2 a; 
sin 2 a; 



1 + cos 2 a; 



a/\ — cos 2 x 



13. 



1 -f- cos 2 a; 
tan (45° + \ x) - tan (45° -\x) 



454 SPHERICAL ASTRONOMY. 

i 

IY. Relative to two arcs A and B. 

1. sin (A + B) = sin -4 . cos B -f cos A .sin J?. 

2. sin (A — 2?) = sin J. . cos 2? — cos A . sin 2?. 

3. cos (A -\- B) = cos A . cos 2? — sin A . sin 2?. 

4. cos (-4 — 2?) = cos A . cos B + sin ^4 . sin B. 

. . _. tan ^4. + tan 2? 

5. tan (A + B) = - T . = . 

v ' 1 — tan .4 . tan B 

, . _ x tan ^4 — tan 2? 

6. tan (A-B) = — 



tan .4 . tan 2? 

7. rin(45°=fc-B)) cos^dbsini? 

8. cos (45° zp B) 



9. tan (45° do B) 



1 db tan i? 

1 zp tan B 



10. to .(45«±^=i*$4. 

x J 1 qp sin 2? 



11. tan (45° =b i .B) = 
sin (^4 + B) 



cos B 



12 



13. 



14. 



16. 



1 do sin .5 

cos 2? 1 =j= sin B * 

tan J. + tan B cot 2? -+- c °t A 
tan -4 — tan B cot B — cot -4 

cot B — tan A cot A — tan 2? 
cos (A — B) cot B -f tan -4 ~~ cot A + tan 2? 

sin J. + sin ^ _ tan £ (^4 + #) 
sin A — sin 2? ~ tan £ (J. — B) ' 



sin (4 - B) 
cos (^4 + jB) 



cos B + cos ^4 cot J (-4 -f- -5) 
cob B — cos J. "~ tan j \4 — 2?) 



> 



[conftftuerf. 



TRIGONOMETRICAL FORMULAE. 455 

IV. continued. Relative to two arcs A and B. 

16. sin A . cos B = \ sin (A + B) + I sin (A — B). 

IV. cos A . sin B =% sin £4 -f B) — J sin (J — B). 

18. sin ^4 . sin B =\cw{A— B) — \ cos (A + B). 

19. cos A . cos i? = i cos (^4 -f i?) + J cos (^1 — i?). 

20. sin A + sin i? = 2 sin J (^4 + B) . cos J {A — B). 

21. cos A + cos B = 2 cos i (A + ^) . cos A (.4 — i?). 

sin (A + £) 

22. tan A -f tan £ = \ '- . 

cos A . cos i* 

sin (A + i?) 

23. COt A + COt £ = -r-\ : ^. 

sin A . sin B 

24. sin .4 — sin B = 2 sin l (^ — ^) . cos | (^ + 5). 

25. cos B — cos A = 2 sin | (A — B) . sin J (.4 + 2?). 
sin (A — B) 



26. tan A — tan 2? 



cos ^4 . cos -5 * 



sio (^4 - ^) 

27. cot B- cot ^4 =-7— \ — : — £. 

sin A . &m B 

28. sin 2 ^4 - sin 2 B ) 

y = sin (J. - .#) . sin (^4 + .#). 

29. cos 2 £ — cos 2 A ) 

30. cos 1 J. — sin 2 B = cos (A — B) . cos (A + .#). 

31. tfi-tfjwti^^, 

cos a A . cos 2 i/ 

sin (A — B) . sin (J. + £) 

32. cot 2 B - cot 2 ^4 = — i :-=-£ — , ,\ -—!■ . 

sin' A . sin^ i> 



456 SPHERICAL ASTRONOMY. 

Y. Differences of trigonometrical lines. 

1. A sin # = -f 2 sin -J A x . cos (x -f J A «). 

2. A cos a; = — 2 sin J A # . sin (x + J A «). 
sin A x 



I. A tan x = -f- 



4. A cot a; = — 



cos a; . cos (x + A x) 
sin A # 



sin x . sin (a? + A #) 

5. A sin 2 x = + sin A x . sin (2 # + A a;). 

6. A cos 2 x = — sin A x . sin (2 x -f A #). 
2 sin A a; . sin (2 # + A a-) 



1. A tan 2 a; = + 



cos 2 x . cos 2 (a; -f A x) 
sin A a; . sin (2 x -4- A x) 



8. A cot 2 x = - 

sin 4 x . sin** (# + A x) 



VI. Differentials of trigonometrical lines. 

1. d sin x = -f- d x . cos #. 

2. d cos a; = — d x . sin a?. 



da? 
8. d tan x = H — 

cos a; 

4. d cot a; = r-r— , 



sin' x 

,2 



5. d sin 2 x = + 2 d x . sin a? . cos r. 

6. d cos 2 a; = — 2 d x . sin a; . cos x. 

2 d x . tan a; / 

1. d tan 2 a; = H - 9 . * r% ~ fi L / 



8. i cot 2 x = - 



cos' a; 
2 d ar . cot a; 



TRIGONOMETRICAL FORMULAE. 457 

YII. General analytical expressions for the sides and angles 
of any spherical triangle. 

1. cos S = cos A . sin S r . sin S" + cos S r . cos S" 

2. cos S r = cos A . sin S" . sin S + cos S" . cos S. 

3. cos £" = cos A" . sin S . sin £' + cos S . cos £'. 

4. cos ^1 = cos S . sin -4' . sin A" — cos A' . cos A". 

5. cos A' = cos #' . sin A" . sin .4 — cos -4" . cos A. 

6. cos -4" = cos S" . sin ^4 . sin A' — cos A . cos A'. 

7. cos 5 . cos A' = cot S" . sin # — sin -4' . cot -4". 

8. cos S f . cos A" = cot S . sin #' — sin A" . cot A. 

9. cos £" . cos A = cot 5' . sin S" — sin A . cot ^4'. 



sin A sin .4' sin ^4' 
10. 



sin S sin £' sin £" ' 

11. sin \ (S f + S) : sin \ (S f - S) : : cot ± 4" : tan ± (^4' - A). 

12. cosi (£' + s) : cosj (£' - JS) : : cot i A" : tan 1 (4' + A). 

13. sin £ (A' + ^4) : sin | (.4' - .4) • : tan i £" : tan i (£' - £). 

14. cos i (.4' + ^4) : cos * (A f - A) : : tan A £" : tan £ (£' + £). 

In these formulae A, A', A", denote the several angles of the triangle ; 
and S, S', S", the sides opposite those angles respectively. For the 
more convenient computation of the formulae Nos. 1-9, certain auxiliary 
angles are introduced, which will be alluded to in the formulae for the 
solution of the several cases of oblique-angled spherical triang-les. 



458 



SPHERICAL ASTRONOMY. 



YIII. Solutions of the cases of right-angled spherical triangles. 



Given. Required. Solution. 

side op. giv. ang. 1. s ; n x = sin h . sin a. 

side adj. giv. ang. 2. tan x = tan h . cos a. 

the other angle. 3. cot x = cos k . tan a. 



Hypothen 

and 
an angle. 



Hypothen. 

and 

a side. 



the other side. 



4. cos x = 



cos h 

cos * ' 



ang. adj. giv. side. 5. cos x = tan s . cot h, 

„ . sin 5 

ang. op. giv. side. 6. sin x == - — - . 



the hypothen. 1. sin # 

A side and 

the. angle •( the other side. 8. sin x ■== tan s . cot a )- 1 

opposite. 

the other angle. 9. sin # = 



cos 



. . , , r the hypothen. 10. cot x = cos a . cot s. 

the angle <{ the other side. 11. tan # = tan a . sin s. 

J ' I the other angle. 12. cos x 



sin a . cos s. 



The two ( the hypothen. 13. cos x = rectang. cos of the given sides. 
( an angle. 14. cot x = sin adj. side X cot op. side. 

the hypothen. 15. cos x = rectang. cot of the giv. angles 



The two 
angles. ] a s i de> 



16. cos x = 



cos opp. ang. 
sin adj. ang. ' 



In these formulae, x denotes the quantity sought. 

a = the given angle. 
s = the given si< 7 .e. 
h = the hypothenuse. 



TRIGONOMETRICAL FORMULAE. 459 



IX. Solutions of the cases of oblique- angled spherical 

triangles. 

Given, Two sides and an angle opposite one of them. 

Required, 1°. The angle opposite the other given side. 

sin side op. ang. sought X sin giv. ang. 

sin x = ; — —, : \ • 

sin side oppos. given angle 

Required, 2°. The angle included between the given sides. 

cot a' = tan giv. ang. x cos adj. side, 

cos a' x tan side adj. giv. ang. 

cos a" = ^ — ■ 6 , * - , 

tan side op. given angle 

x = (o' ± a"). 

Required, 3°. The third side. 

tan a' = cos giv. ang. x tan adj. side, 

„ _ cos a' X cos side op. giv. ang. 
cos side adj. given angle ' 

x = {a! ± a"). 

In these formulae, x denotes the quantity sought : a' and a" are 
auxiliary angles introduced for the purpose of facilitating the compu- 
tations. 

The angle sought in formula 1 is, in certain cases, ambiguous. In 
the formulae 2 and 3, when the angles opposite to the given sides are 
of the same species, we must take the upper sign ; on the contrary, the 
lower sign. The whole of these formulae therefore are, in certain cases, 
ambiguous. 

[continued. 



460 SPHERICAL ASTRONOMY. 



IX. continued. Solutions of the cases of oblique-angled 
spherical triangles. 

Given, Two angles and a side opposite one of them. 
Required, 4°. The side opposite the other given angle. 



sin ang. op. side sought X sin giv. side 



sin x = 

sin ang. op. given side 



Required, 5°. The side included between the given angles. 

tan a' = tan giv. side X cos ang. adj. giv. side, 

„ sin a' X tan ang. adj. giv. side 

sin a ^^ ~- — ——~ ' 1 . -. , 

tan ang. op. given side 

x =(a'± a"). 



Required, 6°. The third angle. 




cot a' = cos given side X tan adj. angle, 

„ sin a' X cos ang. op. giv. side 

cos ang. adj. given side ' 

x = (a' ± a"). 



In these formulae, x denotes the quantity sought: a' and a" are 
auxiliary angles introduced for the purpose of facilitating the compu- 
tations. 

The side sought in formula 4 is, in certain cases, ambiguous. In 
the formulae 5 and 6, when the sides opposite the given angles are of 
the same species, we must take the upper sign; on the contrary, the 
lower sign. The whole of these formulae therefore are, in certain cases, 
ambiguous. 

[continued. 



TRIGONOMETRICAL FORMULAE. 461 



IX. continued. Solutions of the cases of oblique-angled 
spherical triangles. 

Given, Two sides and the included angle. 

Required, 7°. One of the other angles. 

tan a' = cos given angle X tan given side, 
a" = the base — a' 

, sin a r 
tan x = tan given angle X rr. 

° sin a 

.' • In this formula, the given side is assumed to be the 

side opposite the angle sought : the other known side 
is called the base. 



Required, 8°. The third side. 



\J 



tan a' = cos given angle X tan given side, 
a" = the base ~ a\ 

nna n." 

cos x = cos given side X 



cos a 



In this formula, either of the given sides may be ag» 
slimed as the base; and the other as the given side. 



In these formulae, x denotes the quantity sought : a' and a" are 
auxiliary angles introduced for the purpose of facilitating the compu- 
tations. 

If the side sought in formula 8 be small, the formula may not give 
the value to a sufficient degree of accuracy ; and some other mode must 
be adopted for obtaining the correct value* 

[continued. 



462 SPHERICAL ASTRONOMY. 



IX. continued. Solutions of the cases of obligue-2Liig\Q<\ 
spherical triangles. 

Given, A side and the two adjacent angles. 



JRequired, 9°. One of the other sides. 

cot a' = tan given angle X cos given side, 
a" = the vertical angle ~ a', 



. . cos a' 
tan x = tan given side X -, 



\ 



In this formula, the angle, opposite the side sought, 
is assumed as the given angle : the other known 
angle is called the vertical angle. 



r\ 



Required, 10°. The third angle. 

cot a' = tan given angle X cos given side, 

a" = the vertical angle — a', \^_ ¥t ^ 

sin a" 

cos x = cos given angle X r « 

In this formula, either of the given angles may be 
assumed as the vertical angle ; and the other as the 
given angle. 



In these formulae, * denotes the quantity sought: a 1 and a" are 
auxiliary angles introduced tor the purpose of facilitating the compu- 
tation*. 

If the angle sought in formula 10 be small, the formula may not give 
the value to a sufficient degree of accuracy; and some other mode 
must be adopted for obtaining the correct value 

[continued. 



TRIGONOMETRICAL FORMULAE. 463 

IX. continued. Solutions of the cases of oblique-angled 
spherical triangles. 

Given, The three sides. 

Required, 11°. An angle. 

. (A + B + Ji 



sin I - B ) X sin I 

wi.,^ * § £ I 

sin 5 . 

(A + B + C\ . I 



2 

Sin' ^ X = r-^ : -^ 

£ sin B . sin C 



. (A + B + C\ /A + B + C . 
sin I I X sin I A 



cos 8 1 X 



sin i? . sin C 

In these formulae, A, B, G are the three sides of the 
triangle ; and A is assumed as the fcide opposite to the 
angle required. 



Given, The three angles. 

Required, 12°. A side. 

(a + b + c\ /a + b + c \ 

A — 2~ ) *<*»{— T—~ a ) 



sin 8 ^ x = — 



cos 

\ 2 

sin 6 . sin c 



cos^— 6j x cos (— - cj 

COS 8 * # = r-h ; -. 

- sin 6 . sin c 

In these i> ivunulse, a, o, c are the three angles of the 
triangle ; and a is assumed as the angle opposite to 
the side required. 

In these formulae, x denotes the quantity sought. The formulae, which 
are resolved by the cosine, are used only when the angle or side x is 
mi all. 



464 



SPHERICAL ASTRONOMY. 



X. Trigonometrical series. 



1. sin x = x — - — - + 



2.3 2.3.4.5 



— &c. 



2. cos x — 1 — .-— - + - — - — - — 



2 '2.3.4 2.3.4.5.6 



-f- Ac 



, x 1 , 2 a; 5 , 11 x 7 
3 . tan s = a: + y + — + ___ + te 






2 * a? 2 s 5 

4. COt £ = — £r— - ; — — = — — &C. 

x 3 3 2 .5 3 3 .5.7 

\ 



5. ver-sin x = —- — 



+ 



2 2 . 3 . 4 2 . 3 . 4 . 5 . 6 



x = sin x + 



sin 3 x 1 . 3 sin 5 a; 
2.3 2.4.5 



+ &c. 



1. 



ir cos 3 a; 1.3 cos 5 x 

x = - — cos a; — - — — - — <fec. 

2 2.3 2.4.5 



8. x = tan a; — J tan 3 a; + J tan 5 x — &c. 

V 

In the series No. 7, * denotes the periphery of the circle, or 
8.14159265. V 



TRIGONOMETRICAL FORMULAE. 



465 



XL Multiple arcs. 





sin = 0, 

sin x = sin x, 
sin 2 x = 2 sin a? . cos a?, 
sin 3 a; = 2 sin x . cos 2 a; + sin #, 
sin 4 x = 2 sin a? . cos 3 x + sin 2 a?, 

<fec. &c. &c. 

cos = 1, 

cos x = cos #, 
cos 2 x = 2 cos a; . cos x — 1, 
cos 3 x = 2 cos a; . cos 2 x — cos a;, 
cos 4 a; = 2 cos a; . cos 3 x — cos 2 * 

&c. &c. &c. 



tan a; = tan ar, 



tan 2a? 



tan 3a; = 



tan 4 a; = 



2 tan a; 



1 — 


tan 2 x 1 


tan 


x + tan 2 x 


1 - 


tan a; . tan 2 a?' 


tan 


a; + tan 3 x 



1 — tan x . tan 3 x' 
&c. &c. &c. 






.. 



J 



/ 



/ 







J 






4 



trcfs £-■*"'*- 



-y 




— - ... 






* 

V 



^ 



/////////////// /////// it it* mm 



/////// 



Wyffy Vt6 ftfW? (>tx> yy^p^mrry^ ajyrj 



r-wnii7rnU ^ru/? 



/ ^ to 7^^/ 'Tfr<U? 



7 V 



c*~j /I a^X /^v ^ 



-^-v^ &&<? fat*^ 



<: 



